Logiweb expansion of sup in pyk

Up Help

"File page.tex
\documentclass [fleqn,titlepage]{article}
\setlength {\overfullrule }{1mm}
\input{lgwinclude}

\usepackage{latexsym}

%\setlength{\parindent}{0em}
%\setlength{\parskip}{1ex}

% The font of each Logiweb construct is under tight control except that
% strings are typeset in whatever font is in effect at the time of
% typesetting. This is done to enhance the readability of strings in the
% TeX source generated by Logiweb. The default font for typesetting
% strings is \rm:
\everymath{\rm}

\usepackage{makeidx}
%\usepackage{page} - fjernet 3.5.06
%\makeindex - fjernet 3.5.06
\newcommand{\intro}[1]{\emph{#1}}
\newcommand{\indexintro}[1]{\index{#1}\intro{#1}}
\newlength{\bracketwidth}
\settowidth{\bracketwidth}{$[{}$}
\newcommand{\back}{\protect\makebox[-1.0\bracketwidth]{}}

% tilfoejede pakker
\usepackage{lscape}
\usepackage{multicol}
\usepackage{float}
\usepackage[latin1]{inputenc}
\usepackage[dvips]{graphicx}
\usepackage{verbatim}
\usepackage[danish]{babel}
\usepackage{graphpap}

\usepackage[dvipdfm=true]{hyperref}
\hypersetup{pdfpagemode=none}
\hypersetup{pdfstartpage=1}
\hypersetup{pdfstartview=FitBH}
\hypersetup{pdfpagescrop={120 130 490 730}}
%\hypersetup{pdftitle=}
\hypersetup{colorlinks=false}
\bibliographystyle{plain}

% \tex{something} writes something to page.otx for later inclusion
\newwrite\outex
\newtoks\toktex
\immediate\openout\outex=page.otx
\newcommand{\tex}[1]{\toktex={\item #1}\immediate\write\outex{\the\toktex}}

% \test{something} writes something to page.tst for later inclusion
\newwrite\outest
\immediate\openout\outest=page.tst
\newcommand{\test}[1]{\toktex={\item #1}\immediate\write\outest{\the\toktex}}

% Concerning \catcode`\@=11 : See the TeXbook, Appendix B (page 344).
% \afterheading suppresses indentation once, c.f. latex.ltx.
% \display{something} displays something as a displayed equation except
% that linebreaking is possible and displaymath is not turned on by default.
% The first paragraph after \display{something} is unindented.
% Glue below formulas may be wrong. The definition of \display misses
% something like \addvspace{\belowdisplayskip}.
\catcode`\@=11
\def\afterheading{\@afterheading}
\catcode`\@=12
\newcommand{\display}[1]{\begin{list}{}{\setlength{\leftmargin}{\mathindent}}
\item #1\end{list}
\afterheading}
\newcommand{\statement}[1]{\begin{list}{}{\setlength{\leftmargin}{0mm}}
\item #1\end{list}
\afterheading}

\begin {document}

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

" [ ragged right ] "

SUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUP



" [ math define statement of prop lemma remove or as system Q infer all metavar var a end metavar indeed not0 metavar var a end metavar imply metavar var a end metavar infer metavar var a end metavar end define end math ] "

" [ math define proof of prop lemma remove or as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed not0 metavar var a end metavar imply metavar var a end metavar infer 1rule repetition modus ponens not0 metavar var a end metavar imply metavar var a end metavar conclude not0 metavar var a end metavar imply metavar var a end metavar cut prop lemma auto imply conclude metavar var a end metavar imply metavar var a end metavar cut prop lemma from negations modus ponens metavar var a end metavar imply metavar var a end metavar modus ponens not0 metavar var a end metavar imply metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of prop lemma to negated and as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply not0 metavar var b end metavar infer not0 not0 metavar var a end metavar imply not0 metavar var b end metavar end define end math ] "

" [ math define proof of prop lemma to negated and as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply not0 metavar var b end metavar infer prop lemma add double neg modus ponens metavar var a end metavar imply not0 metavar var b end metavar conclude not0 not0 metavar var a end metavar imply not0 metavar var b end metavar cut 1rule repetition modus ponens not0 not0 metavar var a end metavar imply not0 metavar var b end metavar conclude not0 not0 metavar var a end metavar imply not0 metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of prop lemma to negated and(1) as system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar infer not0 not0 metavar var a end metavar imply not0 metavar var b end metavar end define end math ] "

" [ math define proof of prop lemma to negated and(1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar infer metavar var a end metavar infer prop lemma from contradiction modus ponens metavar var a end metavar modus ponens not0 metavar var a end metavar conclude not0 metavar var b end metavar cut all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var b end metavar indeed not0 metavar var a end metavar infer metavar var a end metavar infer not0 metavar var b end metavar conclude not0 metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar cut not0 metavar var a end metavar infer 1rule mp modus ponens not0 metavar var a end metavar imply metavar var a end metavar imply not0 metavar var b end metavar modus ponens not0 metavar var a end metavar conclude metavar var a end metavar imply not0 metavar var b end metavar cut prop lemma to negated and modus ponens metavar var a end metavar imply not0 metavar var b end metavar conclude not0 not0 metavar var a end metavar imply not0 metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma intro exist helper as system Q infer all metavar var x end metavar indeed all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed meta-sub not0 metavar var a end metavar is not0 metavar var b end metavar where metavar var v1 end metavar is metavar var x end metavar end sub endorse for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar imply not0 metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma intro exist helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed meta-sub not0 metavar var a end metavar is not0 metavar var b end metavar where metavar var v1 end metavar is metavar var x end metavar end sub endorse for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar infer lemma a4 at metavar var x end metavar modus probans meta-sub not0 metavar var a end metavar is not0 metavar var b end metavar where metavar var v1 end metavar is metavar var x end metavar end sub modus ponens for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar conclude not0 metavar var a end metavar cut all metavar var x end metavar indeed all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed meta-sub not0 metavar var a end metavar is not0 metavar var b end metavar where metavar var v1 end metavar is metavar var x end metavar end sub endorse for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar infer not0 metavar var a end metavar conclude meta-sub not0 metavar var a end metavar is not0 metavar var b end metavar where metavar var v1 end metavar is metavar var x end metavar end sub endorse for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar imply not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma intro exist as system Q infer all metavar var x end metavar indeed all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed meta-sub not0 metavar var a end metavar is not0 metavar var b end metavar where metavar var v1 end metavar is metavar var x end metavar end sub endorse metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar end define end math ] "

" [ math define proof of pred lemma intro exist as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed meta-sub not0 metavar var a end metavar is not0 metavar var b end metavar where metavar var v1 end metavar is metavar var x end metavar end sub endorse pred lemma intro exist helper at metavar var x end metavar modus probans meta-sub not0 metavar var a end metavar is not0 metavar var b end metavar where metavar var v1 end metavar is metavar var x end metavar end sub conclude for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar imply not0 metavar var a end metavar cut metavar var a end metavar infer prop lemma add double neg modus ponens metavar var a end metavar conclude not0 not0 metavar var a end metavar cut prop lemma mt modus ponens for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar imply not0 metavar var a end metavar modus ponens not0 not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar cut 1rule repetition modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma exist mp as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer metavar var b end metavar end define end math ] "

" [ math define proof of pred lemma exist mp as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer not0 metavar var b end metavar infer prop lemma mt modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar cut 1rule gen modus ponens not0 metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar cut 1rule repetition modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar cut prop lemma from contradiction modus ponens for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude not0 not0 metavar var b end metavar cut all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer not0 metavar var b end metavar infer not0 not0 metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var b end metavar cut metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 metavar var b end metavar imply not0 not0 metavar var b end metavar modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude not0 metavar var b end metavar imply not0 not0 metavar var b end metavar cut prop lemma imply negation modus ponens not0 metavar var b end metavar imply not0 not0 metavar var b end metavar conclude not0 not0 metavar var b end metavar cut prop lemma remove double neg modus ponens not0 not0 metavar var b end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of pred lemma exist mp2 as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer not0 for all objects metavar var v2 end metavar indeed not0 metavar var b end metavar infer metavar var c end metavar end define end math ] "


" [ math define proof of pred lemma exist mp2 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer not0 for all objects metavar var v2 end metavar indeed not0 metavar var b end metavar infer pred lemma exist mp modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude metavar var b end metavar imply metavar var c end metavar cut pred lemma exist mp modus ponens metavar var b end metavar imply metavar var c end metavar modus ponens not0 for all objects metavar var v2 end metavar indeed not0 metavar var b end metavar conclude metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma 2exist mp as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar infer metavar var b end metavar end define end math ] "

" [ math define proof of pred lemma 2exist mp as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar infer pred lemma exist mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar conclude metavar var b end metavar cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar infer metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar cut metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar imply metavar var b end metavar conclude not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar cut pred lemma exist mp modus ponens not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma 2exist mp2 as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var v4 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar infer not0 for all objects metavar var v3 end metavar indeed not0 not0 for all objects metavar var v4 end metavar indeed not0 metavar var b end metavar infer metavar var c end metavar end define end math ] "

" [ math define proof of pred lemma 2exist mp2 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var v4 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar infer not0 for all objects metavar var v3 end metavar indeed not0 not0 for all objects metavar var v4 end metavar indeed not0 metavar var b end metavar infer pred lemma 2exist mp modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 metavar var a end metavar conclude metavar var b end metavar imply metavar var c end metavar cut pred lemma 2exist mp modus ponens metavar var b end metavar imply metavar var c end metavar modus ponens not0 for all objects metavar var v3 end metavar indeed not0 not0 for all objects metavar var v4 end metavar indeed not0 metavar var b end metavar conclude metavar var c end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma allNegated(Imply) as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma allNegated(Imply) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar infer lemma a4 at metavar var x end metavar modus ponens for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar conclude not0 not0 metavar var a end metavar cut prop lemma remove double neg modus ponens not0 not0 metavar var a end metavar conclude metavar var a end metavar cut 1rule gen modus ponens metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed metavar var a end metavar cut all metavar var v1 end metavar indeed all metavar var a end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var a end metavar indeed for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar infer for all objects metavar var v1 end metavar indeed metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar cut prop lemma contrapositive modus ponens for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar cut 1rule repetition modus ponens not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of pred lemma existNegated(Imply) as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma existNegated(Imply) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer 1rule repetition modus ponens not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar cut prop lemma remove double neg modus ponens not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar cut all metavar var v1 end metavar indeed all metavar var a end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var a end metavar indeed not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar infer for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma addAll as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer for all objects metavar var v1 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed metavar var b end metavar end define end math ] "

" [ math define proof of pred lemma addAll as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer for all objects metavar var v1 end metavar indeed metavar var a end metavar infer lemma a4 modus ponens for all objects metavar var v1 end metavar indeed metavar var a end metavar conclude metavar var a end metavar cut 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar conclude metavar var b end metavar cut 1rule gen modus ponens metavar var b end metavar conclude for all objects metavar var v1 end metavar indeed metavar var b end metavar cut all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer for all objects metavar var v1 end metavar indeed metavar var a end metavar infer for all objects metavar var v1 end metavar indeed metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed metavar var b end metavar cut metavar var a end metavar imply metavar var b end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed metavar var b end metavar modus ponens metavar var a end metavar imply metavar var b end metavar conclude for all objects metavar var v1 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma addExist helper1 as system Q infer all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar end define end math ] "

" [ math define proof of pred lemma addExist helper1 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar infer metavar var c end metavar imply metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar infer for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar infer lemma a4 at metavar var y end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar conclude not0 metavar var b end metavar cut prop lemma mt modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens not0 metavar var b end metavar conclude not0 metavar var a end metavar cut prop lemma mt modus ponens metavar var c end metavar imply metavar var a end metavar modus ponens not0 metavar var a end metavar conclude not0 metavar var c end metavar cut 1rule gen modus ponens not0 metavar var c end metavar conclude for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar cut 1rule repetition modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar cut prop lemma from contradiction modus ponens for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar conclude not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar cut all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed 1rule deduction modus ponens all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar infer metavar var c end metavar imply metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar infer for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar infer not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar conclude meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of pred lemma addExist helper2 as system Q infer all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar end define end math ] "

" [ math define proof of pred lemma addExist helper2 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar infer metavar var c end metavar imply metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar infer pred lemma addExist helper1 modus probans meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub conclude metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar cut prop lemma mp3 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var c end metavar imply metavar var a end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar conclude for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar cut prop lemma imply negation modus ponens for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar conclude not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar cut 1rule repetition modus ponens not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar conclude not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar cut all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed 1rule deduction modus ponens all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar infer metavar var c end metavar imply metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar infer not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar conclude meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma addExist as system Q infer all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar infer metavar var c end metavar imply metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar end define end math ] "

" [ math define proof of pred lemma addExist as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar infer metavar var c end metavar imply metavar var a end metavar infer pred lemma addExist helper2 modus probans meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub conclude metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar cut prop lemma mp2 modus ponens metavar var a end metavar imply metavar var b end metavar imply metavar var c end metavar imply metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var c end metavar imply metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var c end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of pred lemma addExist(SimpleAnt) as system Q infer all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar end define end math ] "


" [ math define proof of pred lemma addExist(SimpleAnt) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var y end metavar indeed all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var d end metavar indeed meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub endorse metavar var a end metavar imply metavar var b end metavar infer prop lemma auto imply conclude metavar var a end metavar imply metavar var a end metavar cut pred lemma addExist at metavar var y end metavar modus probans meta-sub not0 metavar var b end metavar is not0 metavar var d end metavar where metavar var v2 end metavar is metavar var y end metavar end sub modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar imply metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var d end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of pred lemma addExist(Simple) as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var b end metavar end define end math ] "

" [ math define proof of pred lemma addExist(Simple) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer prop lemma auto imply conclude metavar var a end metavar imply metavar var a end metavar cut pred lemma addExist at metavar var v2 end metavar modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar imply metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v2 end metavar indeed not0 metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma AEAnegated as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma AEAnegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar infer pred lemma allNegated(Imply) conclude not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma addAll modus ponens not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma existNegated(Imply) conclude not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar cut prop lemma imply transitivity modus ponens not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma addExist(Simple) modus ponens not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma allNegated(Imply) conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar cut prop lemma imply transitivity modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut 1rule mp modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma addEAE as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var b end metavar end define end math ] "

" [ math define proof of pred lemma addEAE as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer pred lemma addExist(Simple) modus ponens metavar var a end metavar imply metavar var b end metavar conclude not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v3 end metavar indeed not0 metavar var b end metavar cut pred lemma addAll modus ponens not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v3 end metavar indeed not0 metavar var b end metavar conclude for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var b end metavar cut pred lemma addExist(Simple) modus ponens for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var b end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma EAE mp as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar infer metavar var b end metavar end define end math ] "

" [ math define proof of pred lemma EAE mp as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar infer lemma a4 at metavar var v2 end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar conclude not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar cut pred lemma exist mp modus ponens metavar var a end metavar imply metavar var b end metavar modus ponens not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar conclude metavar var b end metavar cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed 1rule deduction modus ponens all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed metavar var a end metavar imply metavar var b end metavar infer for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar infer metavar var b end metavar conclude metavar var a end metavar imply metavar var b end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar cut metavar var a end metavar imply metavar var b end metavar infer not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply metavar var b end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar modus ponens metavar var a end metavar imply metavar var b end metavar conclude for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar cut pred lemma exist mp modus ponens for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar imply metavar var b end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 metavar var a end metavar conclude metavar var b end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of pred lemma EEAnegated as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar end define end math ] "

" [ math define proof of pred lemma EEAnegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var v3 end metavar indeed all metavar var a end metavar indeed not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar infer pred lemma allNegated(Imply) conclude not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma addAll modus ponens not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma existNegated(Imply) conclude not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar cut prop lemma imply transitivity modus ponens not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma addAll modus ponens not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut pred lemma existNegated(Imply) conclude not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar cut prop lemma imply transitivity modus ponens not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar modus ponens for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar conclude not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar cut 1rule mp modus ponens not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar modus ponens not0 not0 for all objects metavar var v1 end metavar indeed not0 not0 for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed metavar var a end metavar conclude for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 for all objects metavar var v3 end metavar indeed not0 not0 metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer metavar var x end metavar <= metavar var z end metavar end define end math ] "

" [ math define proof of lemma leqTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer axiom leqTransitivity conclude metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var x end metavar <= metavar var z end metavar cut prop lemma mp2 modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var x end metavar <= metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar <= metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma leqAntisymmetry as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma leqAntisymmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar infer axiom leqAntisymmetry conclude metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar cut prop lemma mp2 modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma leqAddition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer axiom leqAddition conclude metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut 1rule mp modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar infer metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar end define end math ] "

" [ math define proof of lemma leqMultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar infer axiom leqMultiplication conclude 0 <= metavar var z end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar cut prop lemma mp2 modus ponens 0 <= metavar var z end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar modus ponens 0 <= metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma reciprocal as system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar = 0 infer metavar var x end metavar * 1/ metavar var x end metavar = 1 end define end math ] "

" [ math define proof of lemma reciprocal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar = 0 infer axiom reciprocal conclude not0 metavar var x end metavar = 0 imply metavar var x end metavar * 1/ metavar var x end metavar = 1 cut 1rule mp modus ponens not0 metavar var x end metavar = 0 imply metavar var x end metavar * 1/ metavar var x end metavar = 1 modus ponens not0 metavar var x end metavar = 0 conclude metavar var x end metavar * 1/ metavar var x end metavar = 1 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma eqLeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar end define end math ] "

" [ math define proof of lemma eqLeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer axiom eqLeq conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar cut 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma eqAddition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer axiom eqAddition conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end define end math ] "

" [ math define proof of lemma eqMultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer axiom eqMultiplication conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma equality as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar = metavar var z end metavar infer metavar var y end metavar = metavar var z end metavar end define end math ] "

" [ math define proof of lemma equality as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar = metavar var z end metavar infer axiom equality conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var z end metavar imply metavar var y end metavar = metavar var z end metavar cut prop lemma mp2 modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var z end metavar imply metavar var y end metavar = metavar var z end metavar modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar = metavar var z end metavar conclude metavar var y end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqReflexivity as system Q infer all metavar var x end metavar indeed metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma eqReflexivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom leqReflexivity conclude metavar var x end metavar <= metavar var x end metavar cut lemma leqAntisymmetry modus ponens metavar var x end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var x end metavar conclude metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqSymmetry as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma eqSymmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqReflexivity conclude metavar var x end metavar = metavar var x end metavar cut lemma equality modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar = metavar var x end metavar conclude metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar end define end math ] "

" [ math define proof of lemma eqTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut lemma equality modus ponens metavar var y end metavar = metavar var x end metavar modus ponens metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqTransitivity4 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar = metavar var u end metavar end define end math ] "

" [ math define proof of lemma eqTransitivity4 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer lemma eqTransitivity modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar = metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var z end metavar modus ponens metavar var z end metavar = metavar var u end metavar conclude metavar var x end metavar = metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma eqTransitivity5 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var u end metavar = metavar var v end metavar infer metavar var x end metavar = metavar var v end metavar end define end math ] "

" [ math define proof of lemma eqTransitivity5 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var u end metavar = metavar var v end metavar infer lemma eqTransitivity4 modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar = metavar var z end metavar modus ponens metavar var z end metavar = metavar var u end metavar conclude metavar var x end metavar = metavar var u end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var u end metavar modus ponens metavar var u end metavar = metavar var v end metavar conclude metavar var x end metavar = metavar var v end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqTransitivity6 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var u end metavar = metavar var v end metavar infer metavar var v end metavar = metavar var w end metavar infer metavar var x end metavar = metavar var w end metavar end define end math ] "

" [ math define proof of lemma eqTransitivity6 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed all metavar var w end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var u end metavar = metavar var v end metavar infer metavar var v end metavar = metavar var w end metavar infer lemma eqTransitivity5 modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar = metavar var z end metavar modus ponens metavar var z end metavar = metavar var u end metavar modus ponens metavar var u end metavar = metavar var v end metavar conclude metavar var x end metavar = metavar var v end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var v end metavar modus ponens metavar var v end metavar = metavar var w end metavar conclude metavar var x end metavar = metavar var w end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqAdditionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar + metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma eqAdditionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar conclude metavar var z end metavar + metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma eqMultiplicationLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar * metavar var x end metavar = metavar var z end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma eqMultiplicationLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqMultiplication modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var z end metavar modus ponens metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar conclude metavar var z end metavar * metavar var x end metavar = metavar var z end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma plusF(Sym) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end define end math ] "

" [ math define proof of lemma plusF(Sym) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed axiom plusF conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var d end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] + [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma plus0Left as system Q infer all metavar var x end metavar indeed 0 + metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma plus0Left as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom plus0 conclude metavar var x end metavar + 0 = metavar var x end metavar cut axiom plusCommutativity conclude 0 + metavar var x end metavar = metavar var x end metavar + 0 cut lemma eqTransitivity modus ponens 0 + metavar var x end metavar = metavar var x end metavar + 0 modus ponens metavar var x end metavar + 0 = metavar var x end metavar conclude 0 + metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma times1Left as system Q infer all metavar var x end metavar indeed 1 * metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma times1Left as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut axiom timesCommutativity conclude 1 * metavar var x end metavar = metavar var x end metavar * 1 cut lemma eqTransitivity modus ponens 1 * metavar var x end metavar = metavar var x end metavar * 1 modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude 1 * metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma induction as system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed meta-sub metavar var b end metavar is metavar var a end metavar where metavar var v1 end metavar is 0 end sub endorse meta-sub metavar var c end metavar is metavar var a end metavar where metavar var v1 end metavar is metavar var v1 end metavar + 1 end sub endorse metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer metavar var a end metavar end define end math ] "

" [ math define proof of lemma induction as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var a end metavar indeed all metavar var b end metavar indeed all metavar var c end metavar indeed meta-sub metavar var b end metavar is metavar var a end metavar where metavar var v1 end metavar is 0 end sub endorse meta-sub metavar var c end metavar is metavar var a end metavar where metavar var v1 end metavar is metavar var v1 end metavar + 1 end sub endorse metavar var b end metavar infer metavar var a end metavar imply metavar var c end metavar infer 1rule gen modus ponens metavar var a end metavar imply metavar var c end metavar conclude for all objects metavar var v1 end metavar indeed metavar var a end metavar imply metavar var c end metavar cut axiom induction modus probans meta-sub metavar var b end metavar is metavar var a end metavar where metavar var v1 end metavar is 0 end sub modus probans meta-sub metavar var c end metavar is metavar var a end metavar where metavar var v1 end metavar is metavar var v1 end metavar + 1 end sub conclude metavar var b end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar imply metavar var c end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar cut prop lemma mp2 modus ponens metavar var b end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar imply metavar var c end metavar imply for all objects metavar var v1 end metavar indeed metavar var a end metavar modus ponens metavar var b end metavar modus ponens for all objects metavar var v1 end metavar indeed metavar var a end metavar imply metavar var c end metavar conclude for all objects metavar var v1 end metavar indeed metavar var a end metavar cut lemma a4 at metavar var v1 end metavar modus ponens for all objects metavar var v1 end metavar indeed metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma toSeries as system Q infer all metavar var fx end metavar indeed all metavar var sy end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var infer for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar end define end math ] "

" [ math define proof of lemma toSeries as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var sy end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var infer for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer 1rule repetition modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut prop lemma join conjuncts modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var conclude not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut 1rule repetition modus ponens not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var conclude not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut prop lemma join conjuncts modus ponens not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut 1rule repetition modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromSeries as system Q infer all metavar var fx end metavar indeed all metavar var sy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar end define end math ] "


" [ math define proof of lemma fromSeries as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var sy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer 1rule repetition modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut 1rule repetition modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut 1rule repetition modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut 1rule repetition modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma neqSymmetry as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma neqSymmetry as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar = metavar var x end metavar infer lemma eqSymmetry modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar = metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar cut not0 metavar var x end metavar = metavar var y end metavar infer prop lemma mt modus ponens metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveNonzero as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 metavar var x end metavar = 0 end define end math ] "

" [ math define proof of lemma positiveNonzero as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer 1rule repetition modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar cut prop lemma second conjunct modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 = metavar var x end metavar cut lemma neqSymmetry modus ponens not0 0 = metavar var x end metavar conclude not0 metavar var x end metavar = 0 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma subNeqLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var z end metavar infer not0 metavar var y end metavar = metavar var z end metavar end define end math ] "

" [ math define proof of lemma subNeqLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var z end metavar infer axiom equality conclude metavar var y end metavar = metavar var x end metavar imply metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut 1rule mp modus ponens metavar var y end metavar = metavar var x end metavar imply metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar cut prop lemma contrapositive modus ponens metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar = metavar var z end metavar imply not0 metavar var y end metavar = metavar var z end metavar cut 1rule mp modus ponens not0 metavar var x end metavar = metavar var z end metavar imply not0 metavar var y end metavar = metavar var z end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var y end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma inPair(1) as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sx end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end define end math ] "

" [ math define proof of lemma inPair(1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed lemma eqReflexivity conclude metavar var sx end metavar = metavar var sx end metavar cut prop lemma weaken or second modus ponens metavar var sx end metavar = metavar var sx end metavar conclude not0 metavar var sx end metavar = metavar var sx end metavar imply metavar var sx end metavar = metavar var sy end metavar cut lemma formula2pair modus ponens not0 metavar var sx end metavar = metavar var sx end metavar imply metavar var sx end metavar = metavar var sy end metavar conclude metavar var sx end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma inPair(2) as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sy end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end define end math ] "

" [ math define proof of lemma inPair(2) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed lemma eqReflexivity conclude metavar var sy end metavar = metavar var sy end metavar cut prop lemma weaken or first modus ponens metavar var sy end metavar = metavar var sy end metavar conclude not0 metavar var sy end metavar = metavar var sx end metavar imply metavar var sy end metavar = metavar var sy end metavar cut lemma formula2pair modus ponens not0 metavar var sy end metavar = metavar var sx end metavar imply metavar var sy end metavar = metavar var sy end metavar conclude metavar var sy end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma fromSingleton as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair infer metavar var sx end metavar = metavar var sy end metavar end define end math ] "

" [ math define proof of lemma fromSingleton as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair infer 1rule repetition modus ponens metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair conclude metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair cut lemma pair2formula modus ponens metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair conclude not0 metavar var sx end metavar = metavar var sy end metavar imply metavar var sx end metavar = metavar var sy end metavar cut prop lemma remove or modus ponens not0 metavar var sx end metavar = metavar var sy end metavar imply metavar var sx end metavar = metavar var sy end metavar conclude metavar var sx end metavar = metavar var sy end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma toSingleton as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sx end metavar = metavar var sy end metavar infer metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair end define end math ] "

" [ math define proof of lemma toSingleton as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sx end metavar = metavar var sy end metavar infer prop lemma weaken or first modus ponens metavar var sx end metavar = metavar var sy end metavar conclude not0 metavar var sx end metavar = metavar var sy end metavar imply metavar var sx end metavar = metavar var sy end metavar cut lemma formula2pair modus ponens not0 metavar var sx end metavar = metavar var sy end metavar imply metavar var sx end metavar = metavar var sy end metavar conclude metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair cut 1rule repetition modus ponens metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair conclude metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromSameSingleton as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair infer metavar var sx end metavar = metavar var sy end metavar end define end math ] "

" [ math define proof of lemma fromSameSingleton as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair infer lemma eqReflexivity conclude metavar var sx end metavar = metavar var sx end metavar cut lemma toSingleton modus ponens metavar var sx end metavar = metavar var sx end metavar conclude metavar var sx end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair cut lemma set equality nec condition(1) modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair modus ponens metavar var sx end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair cut lemma fromSingleton modus ponens metavar var sx end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair conclude metavar var sx end metavar = metavar var sy end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma singletonmembersEqual as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair infer metavar var sx end metavar = metavar var sy end metavar end define end math ] "

" [ math define proof of lemma singletonmembersEqual as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair infer lemma inPair(1) conclude metavar var sx end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair cut lemma set equality nec condition(1) modus ponens zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair modus ponens metavar var sx end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair conclude metavar var sx end metavar in0 zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair cut lemma fromSingleton modus ponens metavar var sx end metavar in0 zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair conclude metavar var sx end metavar = metavar var sz end metavar cut lemma inPair(2) conclude metavar var sy end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair cut lemma set equality nec condition(1) modus ponens zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair modus ponens metavar var sy end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair conclude metavar var sy end metavar in0 zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair cut lemma fromSingleton modus ponens metavar var sy end metavar in0 zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair conclude metavar var sy end metavar = metavar var sz end metavar cut lemma eqSymmetry modus ponens metavar var sy end metavar = metavar var sz end metavar conclude metavar var sz end metavar = metavar var sy end metavar cut lemma eqTransitivity modus ponens metavar var sx end metavar = metavar var sz end metavar modus ponens metavar var sz end metavar = metavar var sy end metavar conclude metavar var sx end metavar = metavar var sy end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma unequalsNotInSingleton as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed not0 metavar var sx end metavar = metavar var sy end metavar infer not0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair end define end math ] "

" [ math define proof of lemma unequalsNotInSingleton as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair infer lemma singletonmembersEqual modus ponens zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair conclude metavar var sx end metavar = metavar var sy end metavar cut all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair infer metavar var sx end metavar = metavar var sy end metavar conclude zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair imply metavar var sx end metavar = metavar var sy end metavar cut not0 metavar var sx end metavar = metavar var sy end metavar infer prop lemma mt modus ponens zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair imply metavar var sx end metavar = metavar var sy end metavar modus ponens not0 metavar var sx end metavar = metavar var sy end metavar conclude not0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma nonsingletonmembersUnequal as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed not0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair infer not0 metavar var sx end metavar = metavar var sy end metavar end define end math ] "

" [ math define proof of lemma nonsingletonmembersUnequal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sx end metavar = metavar var sy end metavar infer lemma eqReflexivity conclude metavar var sx end metavar = metavar var sx end metavar cut lemma same pair modus ponens metavar var sx end metavar = metavar var sx end metavar modus ponens metavar var sx end metavar = metavar var sy end metavar conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair cut 1rule repetition modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair cut lemma eqSymmetry modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair conclude zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair cut all metavar var sx end metavar indeed all metavar var sy end metavar indeed 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sx end metavar = metavar var sy end metavar infer zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude metavar var sx end metavar = metavar var sy end metavar imply zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair cut not0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair infer prop lemma mt modus ponens metavar var sx end metavar = metavar var sy end metavar imply zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair modus ponens not0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude not0 metavar var sx end metavar = metavar var sy end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromOrderedPair as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar end define end math ] "

" [ math define proof of lemma fromOrderedPair as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sx1 end metavar = metavar var sy1 end metavar infer zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer 1rule repetition modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair cut lemma eqReflexivity conclude metavar var sx1 end metavar = metavar var sx1 end metavar cut lemma same pair modus ponens metavar var sx1 end metavar = metavar var sx1 end metavar modus ponens metavar var sx1 end metavar = metavar var sy1 end metavar conclude zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair cut 1rule repetition modus ponens zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair conclude zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair cut lemma eqReflexivity conclude zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair cut lemma same pair modus ponens zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair modus ponens zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair conclude zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair cut 1rule repetition modus ponens zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair cut lemma eqSymmetry modus ponens zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair cut lemma eqTransitivity modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair modus ponens zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair cut lemma inPair(1) conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair cut lemma set equality nec condition(1) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair in0 zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair cut lemma fromSingleton modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair in0 zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair cut lemma fromSameSingleton modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair conclude metavar var sx end metavar = metavar var sx1 end metavar cut lemma eqSymmetry modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair conclude zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair cut lemma same singleton modus ponens zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair end pair cut lemma eqTransitivity modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair modus ponens zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair end pair cut lemma inPair(2) conclude zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair cut lemma set equality nec condition(1) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair end pair modus ponens zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair end pair cut lemma fromSingleton modus ponens zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair end pair conclude zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair cut lemma singletonmembersEqual modus ponens zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude metavar var sx end metavar = metavar var sy end metavar cut lemma eqSymmetry modus ponens metavar var sx end metavar = metavar var sy end metavar conclude metavar var sy end metavar = metavar var sx end metavar cut lemma eqTransitivity4 modus ponens metavar var sy end metavar = metavar var sx end metavar modus ponens metavar var sx end metavar = metavar var sx1 end metavar modus ponens metavar var sx1 end metavar = metavar var sy1 end metavar conclude metavar var sy end metavar = metavar var sy1 end metavar cut prop lemma join conjuncts modus ponens metavar var sx end metavar = metavar var sx1 end metavar modus ponens metavar var sy end metavar = metavar var sy1 end metavar conclude not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed not0 metavar var sx1 end metavar = metavar var sy1 end metavar infer zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer 1rule repetition modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair cut lemma inPair(1) conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair cut lemma set equality nec condition(1) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair in0 zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair cut lemma pair2formula modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair in0 zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude not0 zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair imply zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair cut lemma unequalsNotInSingleton modus ponens not0 metavar var sx1 end metavar = metavar var sy1 end metavar conclude not0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair cut lemma neqSymmetry modus ponens not0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude not0 zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair cut prop lemma negate second disjunct modus ponens not0 zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair imply zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair modus ponens not0 zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair cut lemma fromSameSingleton modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair conclude metavar var sx end metavar = metavar var sx1 end metavar cut lemma inPair(2) conclude zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair in0 zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair cut lemma set equality nec condition(2) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair modus ponens zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair in0 zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair cut lemma pair2formula modus ponens zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude not0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair imply zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair cut prop lemma negate first disjunct modus ponens not0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair imply zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair modus ponens not0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair cut lemma inPair(2) conclude metavar var sy end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair cut lemma set equality nec condition(2) modus ponens zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair modus ponens metavar var sy end metavar in0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair conclude metavar var sy end metavar in0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair cut lemma pair2formula modus ponens metavar var sy end metavar in0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair conclude not0 metavar var sy end metavar = metavar var sx1 end metavar imply metavar var sy end metavar = metavar var sy1 end metavar cut lemma unequalsNotInSingleton modus ponens not0 metavar var sx1 end metavar = metavar var sy1 end metavar conclude not0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair cut lemma subNeqLeft modus ponens zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair modus ponens not0 zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude not0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair cut lemma nonsingletonmembersUnequal modus ponens not0 zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair conclude not0 metavar var sx end metavar = metavar var sy end metavar cut lemma subNeqLeft modus ponens metavar var sx end metavar = metavar var sx1 end metavar modus ponens not0 metavar var sx end metavar = metavar var sy end metavar conclude not0 metavar var sx1 end metavar = metavar var sy end metavar cut lemma neqSymmetry modus ponens not0 metavar var sx1 end metavar = metavar var sy end metavar conclude not0 metavar var sy end metavar = metavar var sx1 end metavar cut prop lemma negate first disjunct modus ponens not0 metavar var sy end metavar = metavar var sx1 end metavar imply metavar var sy end metavar = metavar var sy1 end metavar modus ponens not0 metavar var sy end metavar = metavar var sx1 end metavar conclude metavar var sy end metavar = metavar var sy1 end metavar cut prop lemma join conjuncts modus ponens metavar var sx end metavar = metavar var sx1 end metavar modus ponens metavar var sy end metavar = metavar var sy1 end metavar conclude not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sx1 end metavar = metavar var sy1 end metavar infer zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar conclude metavar var sx1 end metavar = metavar var sy1 end metavar imply zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair imply not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar cut 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed not0 metavar var sx1 end metavar = metavar var sy1 end metavar infer zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar conclude not0 metavar var sx1 end metavar = metavar var sy1 end metavar imply zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair imply not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar cut zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer prop lemma from negations modus ponens metavar var sx1 end metavar = metavar var sy1 end metavar imply zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair imply not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar modus ponens not0 metavar var sx1 end metavar = metavar var sy1 end metavar imply zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair imply not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair imply not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar cut 1rule mp modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair imply not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromOrderedPair(1) as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer metavar var sx end metavar = metavar var sx1 end metavar end define end math ] "

" [ math define proof of lemma fromOrderedPair(1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer lemma fromOrderedPair modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar cut prop lemma first conjunct modus ponens not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar conclude metavar var sx end metavar = metavar var sx1 end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromOrderedPair(2) as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer metavar var sy end metavar = metavar var sy1 end metavar end define end math ] "

" [ math define proof of lemma fromOrderedPair(2) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair infer lemma fromOrderedPair modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar cut prop lemma second conjunct modus ponens not0 metavar var sx end metavar = metavar var sx1 end metavar imply not0 metavar var sy end metavar = metavar var sy1 end metavar conclude metavar var sy end metavar = metavar var sy1 end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma sameMember(2) as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed metavar var sx end metavar = metavar var sy end metavar infer metavar var sy end metavar in0 metavar var sz end metavar infer metavar var sx end metavar in0 metavar var sz end metavar end define end math ] "

" [ math define proof of lemma sameMember(2) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed metavar var sx end metavar = metavar var sy end metavar infer metavar var sy end metavar in0 metavar var sz end metavar infer lemma eqSymmetry modus ponens metavar var sx end metavar = metavar var sy end metavar conclude metavar var sy end metavar = metavar var sx end metavar cut lemma sameMember modus ponens metavar var sy end metavar = metavar var sx end metavar modus ponens metavar var sy end metavar in0 metavar var sz end metavar conclude metavar var sx end metavar in0 metavar var sz end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma toBinaryUnion(1) as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sx end metavar in0 metavar var sy end metavar infer metavar var sx end metavar in0 U( zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair ) end define end math ] "

" [ math define proof of lemma toBinaryUnion(1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sx end metavar in0 metavar var sy end metavar infer lemma inPair(1) conclude metavar var sy end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair cut prop lemma join conjuncts modus ponens metavar var sx end metavar in0 metavar var sy end metavar modus ponens metavar var sy end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair conclude not0 metavar var sx end metavar in0 metavar var sy end metavar imply not0 metavar var sy end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair cut pred lemma intro exist at metavar var sy end metavar modus ponens not0 metavar var sx end metavar in0 metavar var sy end metavar imply not0 metavar var sy end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair conclude not0 for all objects metavar var su end metavar indeed not0 not0 metavar var sx end metavar in0 metavar var su end metavar imply not0 metavar var su end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair cut lemma formula2union modus ponens not0 for all objects metavar var su end metavar indeed not0 not0 metavar var sx end metavar in0 metavar var su end metavar imply not0 metavar var su end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair conclude metavar var sx end metavar in0 U( zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair ) cut 1rule repetition modus ponens metavar var sx end metavar in0 U( zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair ) conclude metavar var sx end metavar in0 U( zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair ) end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma toBinaryUnion(2) as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sx end metavar in0 metavar var sz end metavar infer metavar var sx end metavar in0 U( zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair ) end define end math ] "


" [ math define proof of lemma toBinaryUnion(2) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sx end metavar in0 metavar var sz end metavar infer lemma inPair(2) conclude metavar var sz end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair cut prop lemma join conjuncts modus ponens metavar var sx end metavar in0 metavar var sz end metavar modus ponens metavar var sz end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair conclude not0 metavar var sx end metavar in0 metavar var sz end metavar imply not0 metavar var sz end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair cut pred lemma intro exist at metavar var sz end metavar modus ponens not0 metavar var sx end metavar in0 metavar var sz end metavar imply not0 metavar var sz end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair conclude not0 for all objects metavar var su end metavar indeed not0 not0 metavar var sx end metavar in0 metavar var su end metavar imply not0 metavar var su end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair cut lemma formula2union modus ponens not0 for all objects metavar var su end metavar indeed not0 not0 metavar var sx end metavar in0 metavar var su end metavar imply not0 metavar var su end metavar in0 zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair conclude metavar var sx end metavar in0 U( zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair ) cut 1rule repetition modus ponens metavar var sx end metavar in0 U( zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair ) conclude metavar var sx end metavar in0 U( zermelo pair metavar var sy end metavar comma metavar var sz end metavar end pair ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromOrderedPair(twoLevels) as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sx end metavar in0 metavar var sy end metavar infer metavar var sy end metavar in0 zermelo pair zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair comma zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair end pair infer not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar end define end math ] "

" [ math define proof of lemma fromOrderedPair(twoLevels) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sx end metavar in0 metavar var sy end metavar infer metavar var sy end metavar in0 zermelo pair zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair comma zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair end pair infer 1rule repetition modus ponens metavar var sy end metavar in0 zermelo pair zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair comma zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair end pair conclude metavar var sy end metavar in0 zermelo pair zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair comma zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair end pair cut lemma pair2formula modus ponens metavar var sy end metavar in0 zermelo pair zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair comma zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair end pair conclude not0 metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair imply metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair cut all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair infer metavar var sx end metavar in0 metavar var sy end metavar infer lemma set equality nec condition(1) modus ponens metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair modus ponens metavar var sx end metavar in0 metavar var sy end metavar conclude metavar var sx end metavar in0 zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair cut lemma fromSingleton modus ponens metavar var sx end metavar in0 zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair conclude metavar var sx end metavar = metavar var sz end metavar cut prop lemma weaken or second modus ponens metavar var sx end metavar = metavar var sz end metavar conclude not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar cut all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair infer metavar var sx end metavar in0 metavar var sy end metavar infer lemma set equality nec condition(1) modus ponens metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair modus ponens metavar var sx end metavar in0 metavar var sy end metavar conclude metavar var sx end metavar in0 zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair cut lemma pair2formula modus ponens metavar var sx end metavar in0 zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair conclude not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar cut 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair infer metavar var sx end metavar in0 metavar var sy end metavar infer not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar conclude metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair imply metavar var sx end metavar in0 metavar var sy end metavar imply not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar cut 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair infer metavar var sx end metavar in0 metavar var sy end metavar infer not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar conclude metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair imply metavar var sx end metavar in0 metavar var sy end metavar imply not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar cut prop lemma from disjuncts modus ponens not0 metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair imply metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair modus ponens metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair imply metavar var sx end metavar in0 metavar var sy end metavar imply not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar modus ponens metavar var sy end metavar = zermelo pair metavar var sz end metavar comma metavar var su end metavar end pair imply metavar var sx end metavar in0 metavar var sy end metavar imply not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar conclude metavar var sx end metavar in0 metavar var sy end metavar imply not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar cut 1rule mp modus ponens metavar var sx end metavar in0 metavar var sy end metavar imply not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar modus ponens metavar var sx end metavar in0 metavar var sy end metavar conclude not0 metavar var sx end metavar = metavar var sz end metavar imply metavar var sx end metavar = metavar var su end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma cartProdIsRelation as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end define end math ] "

" [ math define proof of lemma cartProdIsRelation as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer lemma separation2formula modus ponens object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude not0 object var var r1 end var in0 power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power imply not0 not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut prop lemma second conjunct modus ponens not0 object var var r1 end var in0 power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power imply not0 not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut all metavar var sx end metavar indeed all metavar var sy end metavar indeed 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sy end metavar indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule gen modus ponens object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule repetition modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromSubset as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sx end metavar imply object var var s1 end var in0 metavar var sy end metavar infer metavar var sz end metavar in0 metavar var sx end metavar infer metavar var sz end metavar in0 metavar var sy end metavar end define end math ] "

" [ math define proof of lemma fromSubset as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sx end metavar imply object var var s1 end var in0 metavar var sy end metavar infer metavar var sz end metavar in0 metavar var sx end metavar infer 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sx end metavar imply object var var s1 end var in0 metavar var sy end metavar conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sx end metavar imply object var var s1 end var in0 metavar var sy end metavar cut lemma a4 at metavar var sz end metavar modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sx end metavar imply object var var s1 end var in0 metavar var sy end metavar conclude metavar var sz end metavar in0 metavar var sx end metavar imply metavar var sz end metavar in0 metavar var sy end metavar cut 1rule mp modus ponens metavar var sz end metavar in0 metavar var sx end metavar imply metavar var sz end metavar in0 metavar var sy end metavar modus ponens metavar var sz end metavar in0 metavar var sx end metavar conclude metavar var sz end metavar in0 metavar var sy end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subsetIsRelation as system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sy end metavar imply object var var s1 end var in0 metavar var sx end metavar infer for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end define end math ] "

" [ math define proof of lemma subsetIsRelation as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sy end metavar imply object var var s1 end var in0 metavar var sx end metavar infer object var var r1 end var in0 metavar var sy end metavar infer 1rule repetition modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma a4 at object var var r1 end var modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma fromSubset modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sy end metavar imply object var var s1 end var in0 metavar var sx end metavar modus ponens object var var r1 end var in0 metavar var sy end metavar conclude object var var r1 end var in0 metavar var sx end metavar cut 1rule mp modus ponens object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens object var var r1 end var in0 metavar var sx end metavar conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sy end metavar indeed all metavar var sz end metavar indeed all metavar var su end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sy end metavar imply object var var s1 end var in0 metavar var sx end metavar infer object var var r1 end var in0 metavar var sy end metavar infer not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sy end metavar imply object var var s1 end var in0 metavar var sx end metavar imply object var var r1 end var in0 metavar var sy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sy end metavar imply object var var s1 end var in0 metavar var sx end metavar infer prop lemma mp2 modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sy end metavar imply object var var s1 end var in0 metavar var sx end metavar imply object var var r1 end var in0 metavar var sy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sy end metavar imply object var var s1 end var in0 metavar var sx end metavar conclude object var var r1 end var in0 metavar var sy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule gen modus ponens object var var r1 end var in0 metavar var sy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule repetition modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var sy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sz end metavar imply not0 object var var op2 end var in0 metavar var su end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma CPseparationIsRelation as system Q infer all metavar var a end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end define end math ] "

" [ math define proof of lemma CPseparationIsRelation as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set infer lemma separation2formula(1) modus ponens object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set conclude object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut all metavar var a end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set infer object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma cartProdIsRelation conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma subsetIsRelation modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that metavar var a end metavar end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx end metavar imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma toCartProd helper as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end define end math ] "

" [ math define proof of lemma toCartProd helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer object var var s1 end var in0 metavar var sz end metavar infer lemma fromOrderedPair(twoLevels) modus ponens object var var s1 end var in0 metavar var sz end metavar modus ponens metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude not0 object var var s1 end var = metavar var sx end metavar imply object var var s1 end var = metavar var sy end metavar cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer object var var s1 end var = metavar var sx end metavar infer lemma sameMember(2) modus ponens object var var s1 end var = metavar var sx end metavar modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar conclude object var var s1 end var in0 metavar var sx1 end metavar cut lemma toBinaryUnion(1) modus ponens object var var s1 end var in0 metavar var sx1 end metavar conclude object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sy end metavar in0 metavar var sy1 end metavar infer object var var s1 end var = metavar var sy end metavar infer lemma sameMember(2) modus ponens object var var s1 end var = metavar var sy end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar conclude object var var s1 end var in0 metavar var sy1 end metavar cut lemma toBinaryUnion(2) modus ponens object var var s1 end var in0 metavar var sy1 end metavar conclude object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer object var var s1 end var = metavar var sx end metavar infer object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude metavar var sx end metavar in0 metavar var sx1 end metavar imply object var var s1 end var = metavar var sx end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule mp modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar imply object var var s1 end var = metavar var sx end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar conclude object var var s1 end var = metavar var sx end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule deduction modus ponens all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sy end metavar in0 metavar var sy1 end metavar infer object var var s1 end var = metavar var sy end metavar infer object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude metavar var sy end metavar in0 metavar var sy1 end metavar imply object var var s1 end var = metavar var sy end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule mp modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar imply object var var s1 end var = metavar var sy end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar conclude object var var s1 end var = metavar var sy end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut prop lemma from disjuncts modus ponens not0 object var var s1 end var = metavar var sx end metavar imply object var var s1 end var = metavar var sy end metavar modus ponens object var var s1 end var = metavar var sx end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) modus ponens object var var s1 end var = metavar var sy end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer object var var s1 end var in0 metavar var sz end metavar infer object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude metavar var sx end metavar in0 metavar var sx1 end metavar imply metavar var sy end metavar in0 metavar var sy1 end metavar imply metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer prop lemma mp3 modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar imply metavar var sy end metavar in0 metavar var sy1 end metavar imply metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar modus ponens metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule gen modus ponens object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma toCartProd as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end define end math ] "

" [ math define proof of lemma toCartProd as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer lemma toCartProd helper modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar modus ponens object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude for all objects object var var s1 end var indeed object var var s1 end var in0 object var var s1 end var imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut lemma formula2power modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 object var var s1 end var imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power conclude metavar var sx end metavar in0 metavar var sx1 end metavar imply metavar var sy end metavar in0 metavar var sy1 end metavar imply object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power cut metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer prop lemma mp2 modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar imply metavar var sy end metavar in0 metavar var sy1 end metavar imply object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar conclude object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power cut 1rule gen modus ponens object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power conclude for all objects object var var s1 end var indeed object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power cut 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power conclude for all objects object var var s1 end var indeed object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power cut lemma formula2power modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power cut lemma eqReflexivity conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair cut prop lemma join conjuncts modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar conclude not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar cut prop lemma join conjuncts modus ponens not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude not0 not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair cut pred lemma intro exist at metavar var sy end metavar modus ponens not0 not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude not0 for all objects object var var op2 end var indeed not0 not0 not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma object var var op2 end var end pair end pair cut pred lemma intro exist at metavar var sx end metavar modus ponens not0 for all objects object var var op2 end var indeed not0 not0 not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma object var var op2 end var end pair end pair conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule repetition modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma formula2separation modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut 1rule repetition modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma crsIsRelation as system Q infer all metavar var x end metavar indeed for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end define end math ] "

" [ math define proof of lemma crsIsRelation as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set infer 1rule repetition modus ponens object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set conclude object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set cut lemma separation2formula modus ponens object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set conclude not0 object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 not0 for all objects object var var crs1 end var indeed not0 object var var s1 end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair cut prop lemma first conjunct modus ponens not0 object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 not0 for all objects object var var crs1 end var indeed not0 object var var s1 end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair conclude object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set infer object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut 1rule gen modus ponens object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma cartProdIsRelation conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma subsetIsRelation modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma crsIsFunction as system Q infer all metavar var x end metavar indeed not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var end define end math ] "

" [ math define proof of lemma crsIsFunction as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair infer lemma fromOrderedPair modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair conclude not0 object var var f1 end var = object var var crs1 end var imply not0 object var var f2 end var = metavar var x end metavar cut prop lemma second conjunct modus ponens not0 object var var f1 end var = object var var crs1 end var imply not0 object var var f2 end var = metavar var x end metavar conclude object var var f2 end var = metavar var x end metavar cut lemma fromOrderedPair modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair conclude not0 object var var f3 end var = object var var crs1 end var imply not0 object var var f4 end var = metavar var x end metavar cut prop lemma second conjunct modus ponens not0 object var var f3 end var = object var var crs1 end var imply not0 object var var f4 end var = metavar var x end metavar conclude object var var f4 end var = metavar var x end metavar cut lemma eqSymmetry modus ponens object var var f4 end var = metavar var x end metavar conclude metavar var x end metavar = object var var f4 end var cut lemma eqTransitivity modus ponens object var var f2 end var = metavar var x end metavar modus ponens metavar var x end metavar = object var var f4 end var conclude object var var f2 end var = object var var f4 end var cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair infer object var var f2 end var = object var var f4 end var conclude zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair imply object var var f2 end var = object var var f4 end var cut zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set infer object var var f1 end var = object var var f3 end var infer lemma separation2formula modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set conclude not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair cut prop lemma second conjunct modus ponens not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair conclude not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair cut lemma separation2formula modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set conclude not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair cut prop lemma second conjunct modus ponens not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set imply not0 not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair conclude not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair cut pred lemma exist mp2 modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair modus ponens not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair conclude object var var f2 end var = object var var f4 end var cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set infer object var var f1 end var = object var var f3 end var infer object var var f2 end var = object var var f4 end var conclude for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut lemma crsIsRelation conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut prop lemma join conjuncts modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var conclude not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma crsIsTotal as system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed lambda var c dot typeRational0( quote metavar var x end metavar end quote ) endorse metavar var m end metavar in0 N infer zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set end define end math ] "

" [ math define proof of lemma crsIsTotal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed lambda var c dot typeRational0( quote metavar var x end metavar end quote ) endorse metavar var m end metavar in0 N infer axiom rationalType modus probans lambda var c dot typeRational0( quote metavar var x end metavar end quote ) conclude metavar var x end metavar in0 Q cut lemma toCartProd modus ponens metavar var m end metavar in0 N modus ponens metavar var x end metavar in0 Q conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma eqReflexivity conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair cut pred lemma intro exist at metavar var m end metavar modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair conclude not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair cut lemma formula2separation modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 for all objects object var var crs1 end var indeed not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma crsIsSeries as system Q infer all metavar var x end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set end define end math ] "

" [ math define proof of lemma crsIsSeries as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed object var var s1 end var in0 N infer lemma crsIsTotal modus ponens object var var s1 end var in0 N conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma metavar var x end metavar end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set cut pred lemma intro exist at metavar var x end metavar modus ponens object var var s1 end var in0 N conclude not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed object var var s1 end var in0 N infer not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set conclude object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set cut 1rule gen modus ponens object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set cut lemma crsIsFunction conclude not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut prop lemma join conjuncts modus ponens not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma crsLookup as system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed metavar var m end metavar in0 N infer [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set ; metavar var m end metavar ] = metavar var x end metavar end define end math ] "

" [ math define proof of lemma crsLookup as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var x end metavar indeed metavar var m end metavar in0 N infer lemma crsIsSeries conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set cut lemma memberOfSeries modus ponens metavar var m end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set cut lemma crsIsTotal modus ponens metavar var m end metavar in0 N conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set cut lemma eqReflexivity conclude metavar var m end metavar = metavar var m end metavar cut lemma uniqueMember modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma metavar var x end metavar end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set modus ponens metavar var m end metavar = metavar var m end metavar conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma metavar var x end metavar end pair end pair end set ; metavar var m end metavar ] = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma 0f as system Q infer all metavar var m end metavar indeed metavar var m end metavar in0 N infer [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 end define end math ] "

" [ math define proof of lemma 0f as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed metavar var m end metavar in0 N infer lemma crsLookup modus ponens metavar var m end metavar in0 N conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 cut 1rule repetition modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma 1f as system Q infer all metavar var m end metavar indeed metavar var m end metavar in0 N infer [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] = 1 end define end math ] "



" [ math define proof of lemma 1f as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed metavar var m end metavar in0 N infer lemma crsLookup modus ponens metavar var m end metavar in0 N conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] = 1 cut 1rule repetition modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] = 1 conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] = 1 end quote state proof state cache var c end expand end define end math ] "




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






" [ math define statement of lemma distributionOut as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma distributionOut as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed axiom distribution conclude metavar var x end metavar * metavar var y end metavar + metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * metavar var y end metavar + metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar conclude metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma three2twoTerms as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var y end metavar + metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var u end metavar end define end math ] "

" [ math define proof of lemma three2twoTerms as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var y end metavar + metavar var z end metavar = metavar var u end metavar infer lemma eqAdditionLeft modus ponens metavar var y end metavar + metavar var z end metavar = metavar var u end metavar conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var u end metavar cut axiom plusAssociativity conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var y end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var y end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var u end metavar conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma three2threeTerms as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma three2threeTerms as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed axiom plusCommutativity conclude metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar cut lemma three2twoTerms modus ponens metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar cut axiom plusAssociativity conclude metavar var x end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar modus ponens metavar var x end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + metavar var z end metavar = metavar var x end metavar + metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma three2twoFactors as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var y end metavar * metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var u end metavar end define end math ] "

" [ math define proof of lemma three2twoFactors as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var y end metavar * metavar var z end metavar = metavar var u end metavar infer lemma eqMultiplicationLeft modus ponens metavar var y end metavar * metavar var z end metavar = metavar var u end metavar conclude metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var u end metavar cut axiom timesAssociativity conclude metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar * metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar * metavar var z end metavar modus ponens metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var u end metavar conclude metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma x=x+(y-y) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar end define end math ] "

" [ math define proof of lemma x=x+(y-y) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed axiom plus0 conclude metavar var x end metavar + 0 = metavar var x end metavar cut axiom negative conclude metavar var y end metavar + - metavar var y end metavar = 0 cut lemma eqSymmetry modus ponens metavar var y end metavar + - metavar var y end metavar = 0 conclude 0 = metavar var y end metavar + - metavar var y end metavar cut lemma eqAdditionLeft modus ponens 0 = metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + 0 = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma equality modus ponens metavar var x end metavar + 0 = metavar var x end metavar modus ponens metavar var x end metavar + 0 = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x=x+y-y as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar end define end math ] "

" [ math define proof of lemma x=x+y-y as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma x=x+(y-y) conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut axiom plusAssociativity conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x=x*y*(1/y) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar = 0 infer metavar var x end metavar = metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar end define end math ] "

" [ math define proof of lemma x=x*y*(1/y) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar = 0 infer axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut lemma reciprocal modus ponens not0 metavar var y end metavar = 0 conclude metavar var y end metavar * 1/ metavar var y end metavar = 1 cut lemma three2twoFactors modus ponens metavar var y end metavar * 1/ metavar var y end metavar = 1 conclude metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var x end metavar * 1 cut lemma eqTransitivity modus ponens metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var x end metavar * 1 modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar = metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x*0+x=x as system Q infer all metavar var x end metavar indeed metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma x*0+x=x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude metavar var x end metavar = metavar var x end metavar * 1 cut lemma eqAdditionLeft modus ponens metavar var x end metavar = metavar var x end metavar * 1 conclude metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar * 1 cut axiom distribution conclude metavar var x end metavar * 0 + 1 = metavar var x end metavar * 0 + metavar var x end metavar * 1 cut lemma eqSymmetry modus ponens metavar var x end metavar * 0 + 1 = metavar var x end metavar * 0 + metavar var x end metavar * 1 conclude metavar var x end metavar * 0 + metavar var x end metavar * 1 = metavar var x end metavar * 0 + 1 cut lemma plus0Left conclude 0 + 1 = 1 cut lemma eqMultiplicationLeft modus ponens 0 + 1 = 1 conclude metavar var x end metavar * 0 + 1 = metavar var x end metavar * 1 cut lemma eqTransitivity5 modus ponens metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar * 1 modus ponens metavar var x end metavar * 0 + metavar var x end metavar * 1 = metavar var x end metavar * 0 + 1 modus ponens metavar var x end metavar * 0 + 1 = metavar var x end metavar * 1 modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x*0=0 as system Q infer all metavar var x end metavar indeed metavar var x end metavar * 0 = 0 end define end math ] "

" [ math define proof of lemma x*0=0 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma x=x+(y-y) conclude metavar var x end metavar * 0 = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar cut axiom plusAssociativity conclude metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar conclude metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar cut lemma x*0+x=x conclude metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar cut lemma eqAddition modus ponens metavar var x end metavar * 0 + metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar + - metavar var x end metavar cut axiom negative conclude metavar var x end metavar + - metavar var x end metavar = 0 cut lemma eqTransitivity5 modus ponens metavar var x end metavar * 0 = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar modus ponens metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar modus ponens metavar var x end metavar * 0 + metavar var x end metavar + - metavar var x end metavar = metavar var x end metavar + - metavar var x end metavar modus ponens metavar var x end metavar + - metavar var x end metavar = 0 conclude metavar var x end metavar * 0 = 0 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma (-1)*(-1)+(-1)*1=0 as system Q infer - 1 * - 1 + - 1 * 1 = 0 end define end math ] "

" [ math define proof of lemma (-1)*(-1)+(-1)*1=0 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma distributionOut conclude - 1 * - 1 + - 1 * 1 = - 1 * - 1 + 1 cut axiom negative conclude 1 + - 1 = 0 cut axiom plusCommutativity conclude - 1 + 1 = 1 + - 1 cut lemma eqTransitivity modus ponens - 1 + 1 = 1 + - 1 modus ponens 1 + - 1 = 0 conclude - 1 + 1 = 0 cut lemma eqMultiplicationLeft modus ponens - 1 + 1 = 0 conclude - 1 * - 1 + 1 = - 1 * 0 cut lemma x*0=0 conclude - 1 * 0 = 0 cut lemma eqTransitivity4 modus ponens - 1 * - 1 + - 1 * 1 = - 1 * - 1 + 1 modus ponens - 1 * - 1 + 1 = - 1 * 0 modus ponens - 1 * 0 = 0 conclude - 1 * - 1 + - 1 * 1 = 0 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma (-1)*(-1)=1 as system Q infer - 1 * - 1 = 1 end define end math ] "


" [ math define proof of lemma (-1)*(-1)=1 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma x=x+(y-y) conclude - 1 * - 1 = - 1 * - 1 + 1 + - 1 cut axiom times1 conclude - 1 * 1 = - 1 cut lemma eqSymmetry modus ponens - 1 * 1 = - 1 conclude - 1 = - 1 * 1 cut lemma eqAdditionLeft modus ponens - 1 = - 1 * 1 conclude 1 + - 1 = 1 + - 1 * 1 cut lemma eqAdditionLeft modus ponens 1 + - 1 = 1 + - 1 * 1 conclude - 1 * - 1 + 1 + - 1 = - 1 * - 1 + 1 + - 1 * 1 cut axiom plusCommutativity conclude 1 + - 1 * 1 = - 1 * 1 + 1 cut lemma eqAdditionLeft modus ponens 1 + - 1 * 1 = - 1 * 1 + 1 conclude - 1 * - 1 + 1 + - 1 * 1 = - 1 * - 1 + - 1 * 1 + 1 cut axiom plusAssociativity conclude - 1 * - 1 + - 1 * 1 + 1 = - 1 * - 1 + - 1 * 1 + 1 cut lemma eqSymmetry modus ponens - 1 * - 1 + - 1 * 1 + 1 = - 1 * - 1 + - 1 * 1 + 1 conclude - 1 * - 1 + - 1 * 1 + 1 = - 1 * - 1 + - 1 * 1 + 1 cut lemma (-1)*(-1)+(-1)*1=0 conclude - 1 * - 1 + - 1 * 1 = 0 cut lemma eqAddition modus ponens - 1 * - 1 + - 1 * 1 = 0 conclude - 1 * - 1 + - 1 * 1 + 1 = 0 + 1 cut lemma plus0Left conclude 0 + 1 = 1 cut lemma eqTransitivity5 modus ponens - 1 * - 1 = - 1 * - 1 + 1 + - 1 modus ponens - 1 * - 1 + 1 + - 1 = - 1 * - 1 + 1 + - 1 * 1 modus ponens - 1 * - 1 + 1 + - 1 * 1 = - 1 * - 1 + - 1 * 1 + 1 modus ponens - 1 * - 1 + - 1 * 1 + 1 = - 1 * - 1 + - 1 * 1 + 1 conclude - 1 * - 1 = - 1 * - 1 + - 1 * 1 + 1 cut lemma eqTransitivity4 modus ponens - 1 * - 1 = - 1 * - 1 + - 1 * 1 + 1 modus ponens - 1 * - 1 + - 1 * 1 + 1 = 0 + 1 modus ponens 0 + 1 = 1 conclude - 1 * - 1 = 1 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subLeqRight as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar <= metavar var x end metavar infer metavar var z end metavar <= metavar var y end metavar end define end math ] "

" [ math define proof of lemma subLeqRight as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar <= metavar var x end metavar infer lemma eqLeq modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma leqTransitivity modus ponens metavar var z end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var z end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subLeqLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var z end metavar infer metavar var y end metavar <= metavar var z end metavar end define end math ] "

" [ math define proof of lemma subLeqLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var z end metavar infer lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut lemma eqLeq modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma leqTransitivity modus ponens metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma 0<1Helper as system Q infer 1 <= 0 imply 0 <= 1 end define end math ] "

" [ math define proof of lemma 0<1Helper as lambda var c dot lambda var x dot proof expand quote system Q infer 1 <= 0 infer lemma leqAddition modus ponens 1 <= 0 conclude 1 + - 1 <= 0 + - 1 cut axiom negative conclude 1 + - 1 = 0 cut lemma subLeqLeft modus ponens 1 + - 1 = 0 modus ponens 1 + - 1 <= 0 + - 1 conclude 0 <= 0 + - 1 cut lemma plus0Left conclude 0 + - 1 = - 1 cut lemma subLeqRight modus ponens 0 + - 1 = - 1 modus ponens 0 <= 0 + - 1 conclude 0 <= - 1 cut lemma leqMultiplication modus ponens 0 <= - 1 modus ponens 0 <= - 1 conclude 0 * - 1 <= - 1 * - 1 cut lemma x*0=0 conclude - 1 * 0 = 0 cut axiom timesCommutativity conclude 0 * - 1 = - 1 * 0 cut lemma eqTransitivity modus ponens 0 * - 1 = - 1 * 0 modus ponens - 1 * 0 = 0 conclude 0 * - 1 = 0 cut lemma subLeqLeft modus ponens 0 * - 1 = 0 modus ponens 0 * - 1 <= - 1 * - 1 conclude 0 <= - 1 * - 1 cut lemma (-1)*(-1)=1 conclude - 1 * - 1 = 1 cut lemma subLeqRight modus ponens - 1 * - 1 = 1 modus ponens 0 <= - 1 * - 1 conclude 0 <= 1 cut 1rule deduction modus ponens 1 <= 0 infer 0 <= 1 conclude 1 <= 0 imply 0 <= 1 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma 0<1 as system Q infer not0 0 <= 1 imply not0 not0 0 = 1 end define end math ] "

" [ math define proof of lemma 0<1 as lambda var c dot lambda var x dot proof expand quote system Q infer axiom leqTotality conclude not0 0 <= 1 imply 1 <= 0 cut prop lemma auto imply conclude 0 <= 1 imply 0 <= 1 cut lemma 0<1Helper conclude 1 <= 0 imply 0 <= 1 cut prop lemma from disjuncts modus ponens not0 0 <= 1 imply 1 <= 0 modus ponens 0 <= 1 imply 0 <= 1 modus ponens 1 <= 0 imply 0 <= 1 conclude 0 <= 1 cut axiom 0not1 conclude not0 0 = 1 cut prop lemma join conjuncts modus ponens 0 <= 1 modus ponens not0 0 = 1 conclude not0 0 <= 1 imply not0 not0 0 = 1 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma addEquations as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end define end math ] "

" [ math define proof of lemma addEquations as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar = metavar var u end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut lemma eqAdditionLeft modus ponens metavar var z end metavar = metavar var u end metavar conclude metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma positiveToRight(Eq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar + - metavar var y end metavar end define end math ] "

" [ math define proof of lemma positiveToRight(Eq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer lemma eqAddition modus ponens metavar var x end metavar + metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var z end metavar + - metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var z end metavar + - metavar var y end metavar conclude metavar var x end metavar = metavar var z end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveToLeft(Eq)(1 term) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + - metavar var y end metavar = 0 end define end math ] "

" [ math define proof of lemma positiveToLeft(Eq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var y end metavar cut axiom negative conclude metavar var y end metavar + - metavar var y end metavar = 0 cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var y end metavar modus ponens metavar var y end metavar + - metavar var y end metavar = 0 conclude metavar var x end metavar + - metavar var y end metavar = 0 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveToRight(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar infer metavar var x end metavar <= metavar var z end metavar + - metavar var y end metavar end define end math ] "

" [ math define proof of lemma positiveToRight(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar <= metavar var z end metavar + - metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar <= metavar var z end metavar + - metavar var y end metavar conclude metavar var x end metavar <= metavar var z end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma positiveToRight(Leq)(1 term) as system Q infer all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var y end metavar <= metavar var z end metavar infer 0 <= metavar var z end metavar + - metavar var y end metavar end define end math ] "

" [ math define proof of lemma positiveToRight(Leq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var y end metavar <= metavar var z end metavar infer lemma plus0Left conclude 0 + metavar var y end metavar = metavar var y end metavar cut lemma eqSymmetry modus ponens 0 + metavar var y end metavar = metavar var y end metavar conclude metavar var y end metavar = 0 + metavar var y end metavar cut lemma subLeqLeft modus ponens metavar var y end metavar = 0 + metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude 0 + metavar var y end metavar <= metavar var z end metavar cut lemma positiveToRight(Leq) modus ponens 0 + metavar var y end metavar <= metavar var z end metavar conclude 0 <= metavar var z end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma negativeToLeft(Eq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma negativeToLeft(Eq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar infer lemma eqAddition modus ponens metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma three2threeTerms conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subtractEquations as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma subtractEquations as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar infer metavar var z end metavar = metavar var u end metavar infer lemma eqAddition modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar conclude metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar cut lemma plus0Left conclude 0 + metavar var z end metavar = metavar var z end metavar cut lemma eqTransitivity modus ponens 0 + metavar var z end metavar = metavar var z end metavar modus ponens metavar var z end metavar = metavar var u end metavar conclude 0 + metavar var z end metavar = metavar var u end metavar cut lemma positiveToRight(Eq) modus ponens 0 + metavar var z end metavar = metavar var u end metavar conclude 0 = metavar var u end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens 0 = metavar var u end metavar + - metavar var z end metavar conclude metavar var u end metavar + - metavar var z end metavar = 0 cut lemma eqAdditionLeft modus ponens metavar var u end metavar + - metavar var z end metavar = 0 conclude metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar + 0 cut axiom plusAssociativity conclude metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar cut axiom plus0 conclude metavar var y end metavar + 0 = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar + 0 modus ponens metavar var y end metavar + 0 = metavar var y end metavar conclude metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var u end metavar + - metavar var z end metavar = metavar var y end metavar conclude metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subtractEquationsLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar infer metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar = metavar var u end metavar end define end math ] "

" [ math define proof of lemma subtractEquationsLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar infer metavar var x end metavar = metavar var y end metavar infer axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var y end metavar + metavar var u end metavar = metavar var u end metavar + metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar modus ponens metavar var y end metavar + metavar var u end metavar = metavar var u end metavar + metavar var y end metavar conclude metavar var z end metavar + metavar var x end metavar = metavar var u end metavar + metavar var y end metavar cut lemma subtractEquations modus ponens metavar var z end metavar + metavar var x end metavar = metavar var u end metavar + metavar var y end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var z end metavar = metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma eqNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer - metavar var x end metavar = - metavar var y end metavar end define end math ] "

" [ math define proof of lemma eqNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer axiom negative conclude metavar var x end metavar + - metavar var x end metavar = 0 cut axiom negative conclude metavar var y end metavar + - metavar var y end metavar = 0 cut lemma eqSymmetry modus ponens metavar var y end metavar + - metavar var y end metavar = 0 conclude 0 = metavar var y end metavar + - metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var x end metavar = 0 modus ponens 0 = metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + - metavar var x end metavar = metavar var y end metavar + - metavar var y end metavar cut lemma subtractEquationsLeft modus ponens metavar var x end metavar + - metavar var x end metavar = metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude - metavar var x end metavar = - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


(*** NO EQUALITY ***)



" [ math define statement of lemma lessNeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma lessNeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 1rule repetition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma x+y=zBackwards as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var y end metavar + metavar var x end metavar end define end math ] "

" [ math define proof of lemma x+y=zBackwards as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer axiom plusCommutativity conclude metavar var x end metavar + metavar var y end metavar = metavar var y end metavar + metavar var x end metavar cut lemma equality modus ponens metavar var x end metavar + metavar var y end metavar = metavar var z end metavar conclude metavar var z end metavar = metavar var y end metavar + metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x*y=zBackwards as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar = metavar var z end metavar infer metavar var z end metavar = metavar var y end metavar * metavar var x end metavar end define end math ] "

" [ math define proof of lemma x*y=zBackwards as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar = metavar var z end metavar infer axiom timesCommutativity conclude metavar var x end metavar * metavar var y end metavar = metavar var y end metavar * metavar var x end metavar cut lemma equality modus ponens metavar var x end metavar * metavar var y end metavar = metavar var z end metavar conclude metavar var z end metavar = metavar var y end metavar * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma doubleMinus as system Q infer all metavar var x end metavar indeed - - metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma doubleMinus as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom negative conclude - metavar var x end metavar + - - metavar var x end metavar = 0 cut lemma x+y=zBackwards modus ponens - metavar var x end metavar + - - metavar var x end metavar = 0 conclude 0 = - - metavar var x end metavar + - metavar var x end metavar cut lemma negativeToLeft(Eq) modus ponens 0 = - - metavar var x end metavar + - metavar var x end metavar conclude 0 + metavar var x end metavar = - - metavar var x end metavar cut lemma plus0Left conclude 0 + metavar var x end metavar = metavar var x end metavar cut lemma equality modus ponens 0 + metavar var x end metavar = - - metavar var x end metavar modus ponens 0 + metavar var x end metavar = metavar var x end metavar conclude - - metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma neqNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer not0 - metavar var x end metavar = - metavar var y end metavar end define end math ] "

" [ math define proof of lemma neqNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer - metavar var x end metavar = - metavar var y end metavar infer lemma eqNegated modus ponens - metavar var x end metavar = - metavar var y end metavar conclude - - metavar var x end metavar = - - metavar var y end metavar cut lemma doubleMinus conclude - - metavar var x end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens - - metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar = - - metavar var x end metavar cut lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar = - - metavar var x end metavar modus ponens - - metavar var x end metavar = - - metavar var y end metavar modus ponens - - metavar var y end metavar = metavar var y end metavar conclude metavar var x end metavar = metavar var y end metavar cut prop lemma from contradiction modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 - metavar var x end metavar = - metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer - metavar var x end metavar = - metavar var y end metavar infer not0 - metavar var x end metavar = - metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar imply - metavar var x end metavar = - metavar var y end metavar imply not0 - metavar var x end metavar = - metavar var y end metavar cut not0 metavar var x end metavar = metavar var y end metavar infer 1rule mp modus ponens not0 metavar var x end metavar = metavar var y end metavar imply - metavar var x end metavar = - metavar var y end metavar imply not0 - metavar var x end metavar = - metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude - metavar var x end metavar = - metavar var y end metavar imply not0 - metavar var x end metavar = - metavar var y end metavar cut prop lemma imply negation modus ponens - metavar var x end metavar = - metavar var y end metavar imply not0 - metavar var x end metavar = - metavar var y end metavar conclude not0 - metavar var x end metavar = - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subNeqRight as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar = metavar var x end metavar infer not0 metavar var z end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma subNeqRight as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar = metavar var x end metavar infer lemma neqSymmetry modus ponens not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut lemma subNeqLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var y end metavar = metavar var z end metavar cut lemma neqSymmetry modus ponens not0 metavar var y end metavar = metavar var z end metavar conclude not0 metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

% linie 4000

" [ math define statement of lemma neqAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma neqAddition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar infer lemma eqReflexivity conclude metavar var z end metavar = metavar var z end metavar cut lemma subtractEquations modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var z end metavar = metavar var z end metavar conclude metavar var x end metavar = metavar var y end metavar cut prop lemma from contradiction modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar infer not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar imply not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut not0 metavar var x end metavar = metavar var y end metavar infer 1rule mp modus ponens not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar imply not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar imply not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma imply negation modus ponens metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar imply not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma neqMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var z end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end define end math ] "

" [ math define proof of lemma neqMultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var z end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar infer lemma x=x*y*(1/y) modus ponens not0 metavar var z end metavar = 0 conclude metavar var x end metavar = metavar var x end metavar * metavar var z end metavar * 1/ metavar var z end metavar cut lemma eqMultiplication modus ponens metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude metavar var x end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar cut lemma x=x*y*(1/y) modus ponens not0 metavar var z end metavar = 0 conclude metavar var y end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar conclude metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar = metavar var x end metavar * metavar var z end metavar * 1/ metavar var z end metavar modus ponens metavar var x end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar conclude metavar var x end metavar = metavar var y end metavar cut prop lemma from contradiction modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var z end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar infer not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var z end metavar = 0 imply not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar imply not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut not0 metavar var z end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar infer prop lemma mp2 modus ponens not0 metavar var z end metavar = 0 imply not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar imply not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar modus ponens not0 metavar var z end metavar = 0 modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar imply not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut prop lemma imply negation modus ponens metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar imply not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

(*** NEGATIVE ***)




" [ math define statement of lemma uniqueNegative as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = 0 infer metavar var x end metavar + metavar var z end metavar = 0 infer metavar var y end metavar = metavar var z end metavar end define end math ] "

" [ math define proof of lemma uniqueNegative as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = 0 infer metavar var x end metavar + metavar var z end metavar = 0 infer axiom plusCommutativity conclude metavar var y end metavar + metavar var x end metavar = metavar var x end metavar + metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar + metavar var x end metavar = metavar var x end metavar + metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar = 0 conclude metavar var y end metavar + metavar var x end metavar = 0 cut lemma positiveToRight(Eq) modus ponens metavar var y end metavar + metavar var x end metavar = 0 conclude metavar var y end metavar = 0 + - metavar var x end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar = 0 conclude metavar var z end metavar + metavar var x end metavar = 0 cut lemma positiveToRight(Eq) modus ponens metavar var z end metavar + metavar var x end metavar = 0 conclude metavar var z end metavar = 0 + - metavar var x end metavar cut lemma eqSymmetry modus ponens metavar var z end metavar = 0 + - metavar var x end metavar conclude 0 + - metavar var x end metavar = metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar = 0 + - metavar var x end metavar modus ponens 0 + - metavar var x end metavar = metavar var z end metavar conclude metavar var y end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma toNotLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end define end math ] "



" [ math define proof of lemma toNotLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar infer lemma leqAntisymmetry modus ponens metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut prop lemma add double neg modus ponens metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var y end metavar = metavar var x end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar infer not0 not0 metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut metavar var x end metavar <= metavar var y end metavar infer 1rule mp modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut prop lemma add double neg modus ponens metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut 1rule repetition modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut 1rule repetition modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma fromLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar <= metavar var x end metavar end define end math ] "

" [ math define proof of lemma fromLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer lemma toNotLess modus ponens metavar var y end metavar <= metavar var x end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer prop lemma add double neg modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma mt modus ponens metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens not0 not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma fromNotLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar end define end math ] "

" [ math define proof of lemma fromNotLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar infer 1rule repetition modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma remove double neg modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut 1rule mp modus ponens metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma remove double neg modus ponens not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar = metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar = metavar var x end metavar cut lemma eqLeq modus ponens metavar var y end metavar = metavar var x end metavar conclude metavar var y end metavar <= metavar var x end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar infer metavar var y end metavar <= metavar var x end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 1rule mp modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut prop lemma auto imply conclude metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar <= metavar var x end metavar cut axiom leqTotality conclude not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut prop lemma from disjuncts modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar <= metavar var x end metavar conclude metavar var y end metavar <= metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma toLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma toLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar infer lemma fromNotLess modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar <= metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar infer metavar var x end metavar <= metavar var y end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar <= metavar var y end metavar cut not0 metavar var x end metavar <= metavar var y end metavar infer prop lemma negative mt modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar <= metavar var y end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "






(*** LEQ ***)

" [ math define statement of lemma leqLessEq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma leqLessEq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma fromNotLess modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma leqAntisymmetry modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude metavar var x end metavar = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar cut metavar var x end metavar <= metavar var y end metavar infer 1rule mp modus ponens metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar cut 1rule repetition modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma lessLeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar end define end math ] "

" [ math define proof of lemma lessLeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 1rule repetition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma first conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma from leqGeq as system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar imply metavar var a end metavar infer metavar var y end metavar <= metavar var x end metavar imply metavar var a end metavar infer metavar var a end metavar end define end math ] "

" [ math define proof of lemma from leqGeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar imply metavar var a end metavar infer metavar var y end metavar <= metavar var x end metavar imply metavar var a end metavar infer axiom leqTotality conclude not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut prop lemma from disjuncts modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar imply metavar var a end metavar modus ponens metavar var y end metavar <= metavar var x end metavar imply metavar var a end metavar conclude metavar var a end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma subLessRight as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar infer not0 metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var z end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma subLessRight as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar infer 1rule repetition modus ponens not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar cut prop lemma first conjunct modus ponens not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar conclude metavar var z end metavar <= metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var z end metavar <= metavar var x end metavar conclude metavar var z end metavar <= metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var z end metavar = metavar var x end metavar cut lemma subNeqRight modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var z end metavar = metavar var y end metavar cut prop lemma join conjuncts modus ponens metavar var z end metavar <= metavar var y end metavar modus ponens not0 metavar var z end metavar = metavar var y end metavar conclude not0 metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subLessLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar end define end math ] "

" [ math define proof of lemma subLessLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar infer 1rule repetition modus ponens not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar cut prop lemma first conjunct modus ponens not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar conclude metavar var x end metavar <= metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar cut prop lemma second conjunct modus ponens not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut lemma subNeqLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var y end metavar = metavar var z end metavar cut prop lemma join conjuncts modus ponens metavar var y end metavar <= metavar var z end metavar modus ponens not0 metavar var y end metavar = metavar var z end metavar conclude not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqLessTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end define end math ] "

" [ math define proof of lemma leqLessTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar infer prop lemma first conjunct modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar cut prop lemma second conjunct modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude not0 metavar var y end metavar = metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar = metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var z end metavar <= metavar var y end metavar cut lemma leqAntisymmetry modus ponens metavar var y end metavar <= metavar var z end metavar modus ponens metavar var z end metavar <= metavar var y end metavar conclude metavar var y end metavar = metavar var z end metavar cut prop lemma from contradiction modus ponens metavar var y end metavar = metavar var z end metavar modus ponens not0 metavar var y end metavar = metavar var z end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar infer not0 metavar var x end metavar = metavar var z end metavar conclude metavar var x end metavar <= metavar var y end metavar imply not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar imply not0 metavar var x end metavar = metavar var z end metavar cut metavar var x end metavar <= metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer prop lemma mp2 modus ponens metavar var x end metavar <= metavar var y end metavar imply not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar imply metavar var x end metavar = metavar var z end metavar imply not0 metavar var x end metavar = metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar = metavar var z end metavar imply not0 metavar var x end metavar = metavar var z end metavar cut prop lemma imply negation modus ponens metavar var x end metavar = metavar var z end metavar imply not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut prop lemma first conjunct modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar cut lemma leqTransitivity modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar <= metavar var z end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var z end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessAddition as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma lessAddition as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessLeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma lessNeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar cut lemma neqAddition modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessAdditionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var z end metavar + metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma lessAdditionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessAddition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar cut lemma subLessLeft modus ponens metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar modus ponens not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var z end metavar + metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar cut lemma subLessRight modus ponens metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar modus ponens not0 metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var z end metavar + metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var z end metavar + metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma leqPlus1 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var x end metavar <= metavar var y end metavar + 1 imply not0 not0 metavar var x end metavar = metavar var y end metavar + 1 end define end math ] "

" [ math define proof of lemma leqPlus1 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma lessAdditionLeft modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 metavar var y end metavar + 0 <= metavar var y end metavar + 1 imply not0 not0 metavar var y end metavar + 0 = metavar var y end metavar + 1 cut axiom plus0 conclude metavar var y end metavar + 0 = metavar var y end metavar cut lemma subLessLeft modus ponens metavar var y end metavar + 0 = metavar var y end metavar modus ponens not0 metavar var y end metavar + 0 <= metavar var y end metavar + 1 imply not0 not0 metavar var y end metavar + 0 = metavar var y end metavar + 1 conclude not0 metavar var y end metavar <= metavar var y end metavar + 1 imply not0 not0 metavar var y end metavar = metavar var y end metavar + 1 cut lemma leqLessTransitivity modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens not0 metavar var y end metavar <= metavar var y end metavar + 1 imply not0 not0 metavar var y end metavar = metavar var y end metavar + 1 conclude not0 metavar var x end metavar <= metavar var y end metavar + 1 imply not0 not0 metavar var x end metavar = metavar var y end metavar + 1 end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma leqAdditionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma leqAdditionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar cut axiom plusCommutativity conclude metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var z end metavar = metavar var z end metavar + metavar var x end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + metavar var z end metavar = metavar var z end metavar + metavar var y end metavar modus ponens metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma leqSubtraction as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar end define end math ] "

" [ math define proof of lemma leqSubtraction as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var x end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var x end metavar modus ponens metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar modus ponens metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var x end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqSubtractionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar infer metavar var x end metavar <= metavar var y end metavar end define end math ] "

" [ math define proof of lemma leqSubtractionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar infer axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var z end metavar + metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var z end metavar + metavar var y end metavar cut lemma subLeqRight modus ponens metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma leqSubtraction modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma negativeToLeft(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar infer metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar end define end math ] "

" [ math define proof of lemma negativeToLeft(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma three2threeTerms conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var y end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma negativeToLeft(Leq)(1 term) as system Q infer all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var y end metavar + - metavar var z end metavar infer metavar var z end metavar <= metavar var y end metavar end define end math ] "

" [ math define proof of lemma negativeToLeft(Leq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var y end metavar + - metavar var z end metavar infer lemma negativeToLeft(Leq) modus ponens 0 <= metavar var y end metavar + - metavar var z end metavar conclude 0 + metavar var z end metavar <= metavar var y end metavar cut lemma plus0Left conclude 0 + metavar var z end metavar = metavar var z end metavar cut lemma subLeqLeft modus ponens 0 + metavar var z end metavar = metavar var z end metavar conclude metavar var z end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma positiveToLeft(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar infer metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar end define end math ] "

" [ math define proof of lemma positiveToLeft(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar modus ponens metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma thirdGeq as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var end define end math ] "

" [ math define proof of lemma thirdGeq as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer axiom leqReflexivity conclude metavar var y end metavar <= metavar var y end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 metavar var y end metavar <= metavar var y end metavar cut 1rule exist intro at existential var var c end var at metavar var y end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 metavar var y end metavar <= metavar var y end metavar conclude not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer axiom leqReflexivity conclude metavar var x end metavar <= metavar var x end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var x end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var x end metavar <= metavar var x end metavar imply not0 metavar var y end metavar <= metavar var x end metavar cut 1rule exist intro at existential var var c end var at metavar var x end metavar modus ponens not0 metavar var x end metavar <= metavar var x end metavar imply not0 metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var conclude metavar var x end metavar <= metavar var y end metavar imply not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var conclude metavar var y end metavar <= metavar var x end metavar imply not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var cut axiom leqTotality conclude not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar cut prop lemma from disjuncts modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply metavar var y end metavar <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar imply not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var modus ponens metavar var y end metavar <= metavar var x end metavar imply not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var conclude not0 metavar var x end metavar <= existential var var c end var imply not0 metavar var y end metavar <= existential var var c end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer - metavar var y end metavar <= - metavar var x end metavar end define end math ] "

" [ math define proof of lemma leqNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + - metavar var x end metavar <= metavar var y end metavar + - metavar var x end metavar cut axiom negative conclude metavar var x end metavar + - metavar var x end metavar = 0 cut lemma subLeqLeft modus ponens metavar var x end metavar + - metavar var x end metavar = 0 modus ponens metavar var x end metavar + - metavar var x end metavar <= metavar var y end metavar + - metavar var x end metavar conclude 0 <= metavar var y end metavar + - metavar var x end metavar cut axiom plusCommutativity conclude metavar var y end metavar + - metavar var x end metavar = - metavar var x end metavar + metavar var y end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + - metavar var x end metavar = - metavar var x end metavar + metavar var y end metavar modus ponens 0 <= metavar var y end metavar + - metavar var x end metavar conclude 0 <= - metavar var x end metavar + metavar var y end metavar cut lemma leqAddition modus ponens 0 <= - metavar var x end metavar + metavar var y end metavar conclude 0 + - metavar var y end metavar <= - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma plus0Left conclude 0 + - metavar var y end metavar = - metavar var y end metavar cut lemma x=x+y-y conclude - metavar var x end metavar = - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens - metavar var x end metavar = - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = - metavar var x end metavar cut lemma subLeqLeft modus ponens 0 + - metavar var y end metavar = - metavar var y end metavar modus ponens 0 + - metavar var y end metavar <= - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude - metavar var y end metavar <= - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma subLeqRight modus ponens - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = - metavar var x end metavar modus ponens - metavar var y end metavar <= - metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude - metavar var y end metavar <= - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma addEquations(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar <= metavar var u end metavar infer metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar end define end math ] "

" [ math define proof of lemma addEquations(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar <= metavar var u end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma leqAdditionLeft modus ponens metavar var z end metavar <= metavar var u end metavar conclude metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar cut lemma leqTransitivity modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "



(*** LESS ***)

" [ math define statement of lemma leqNeqLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma leqNeqLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar infer prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut 1rule repetition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma lessMultiplication as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end define end math ] "

" [ math define proof of lemma lessMultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessLeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma lessLeq modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar conclude 0 <= metavar var z end metavar cut lemma leqMultiplication modus ponens 0 <= metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar cut lemma lessNeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar cut lemma lessNeq modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar conclude not0 0 = metavar var z end metavar cut lemma neqSymmetry modus ponens not0 0 = metavar var z end metavar conclude not0 metavar var z end metavar = 0 cut lemma neqMultiplication modus ponens not0 metavar var z end metavar = 0 modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut lemma leqNeqLess modus ponens metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar modus ponens not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessMultiplicationLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar * metavar var x end metavar <= metavar var z end metavar * metavar var y end metavar imply not0 not0 metavar var z end metavar * metavar var x end metavar = metavar var z end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma lessMultiplicationLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessMultiplication modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var x end metavar cut axiom timesCommutativity conclude metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar cut lemma subLessLeft modus ponens metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var x end metavar modus ponens not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var z end metavar * metavar var x end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var z end metavar * metavar var x end metavar = metavar var y end metavar * metavar var z end metavar cut lemma subLessRight modus ponens metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar modus ponens not0 metavar var z end metavar * metavar var x end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var z end metavar * metavar var x end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var z end metavar * metavar var x end metavar <= metavar var z end metavar * metavar var y end metavar imply not0 not0 metavar var z end metavar * metavar var x end metavar = metavar var z end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessDivision as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end define end math ] "



" [ math define proof of lemma lessDivision as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar infer lemma fromLess modus ponens not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar conclude not0 metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar cut axiom leqMultiplication conclude 0 <= metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar cut 1rule mp modus ponens 0 <= metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar modus ponens 0 <= metavar var z end metavar conclude metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar cut prop lemma contrapositive modus ponens metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar conclude not0 metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar imply not0 metavar var y end metavar <= metavar var x end metavar cut 1rule mp modus ponens not0 metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar imply not0 metavar var y end metavar <= metavar var x end metavar modus ponens not0 metavar var y end metavar * metavar var z end metavar <= metavar var x end metavar * metavar var z end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar cut lemma toLess modus ponens not0 metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma lessLeqTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end define end math ] "

" [ math define proof of lemma lessLeqTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer metavar var z end metavar = metavar var x end metavar infer prop lemma first conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut prop lemma second conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar cut lemma subLeqRight modus ponens metavar var z end metavar = metavar var x end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma leqAntisymmetry modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude metavar var x end metavar = metavar var y end metavar cut prop lemma from contradiction modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var z end metavar = metavar var x end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer metavar var z end metavar = metavar var x end metavar infer not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var z end metavar = metavar var x end metavar imply not0 metavar var z end metavar = metavar var x end metavar cut not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var y end metavar <= metavar var z end metavar infer prop lemma mp2 modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var y end metavar <= metavar var z end metavar imply metavar var z end metavar = metavar var x end metavar imply not0 metavar var z end metavar = metavar var x end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var z end metavar = metavar var x end metavar imply not0 metavar var z end metavar = metavar var x end metavar cut prop lemma imply negation modus ponens metavar var z end metavar = metavar var x end metavar imply not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var z end metavar = metavar var x end metavar cut lemma neqSymmetry modus ponens not0 metavar var z end metavar = metavar var x end metavar conclude not0 metavar var x end metavar = metavar var z end metavar cut prop lemma first conjunct modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma leqTransitivity modus ponens metavar var x end metavar <= metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar <= metavar var z end metavar cut prop lemma join conjuncts modus ponens metavar var x end metavar <= metavar var z end metavar modus ponens not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessTransitivity as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end define end math ] "

" [ math define proof of lemma lessTransitivity as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar infer prop lemma first conjunct modus ponens not0 metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var z end metavar conclude metavar var y end metavar <= metavar var z end metavar cut lemma lessLeqTransitivity modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar <= metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma addEquations(Less) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar infer not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end define end math ] "

" [ math define proof of lemma addEquations(Less) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar infer lemma lessAddition modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar cut lemma lessAdditionLeft modus ponens not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar conclude not0 metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar cut lemma lessTransitivity modus ponens not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessTotality as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end define end math ] "


" [ math define proof of lemma lessTotality as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar infer lemma fromNotLess modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma neqSymmetry modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var y end metavar = metavar var x end metavar cut lemma leqNeqLess modus ponens metavar var y end metavar <= metavar var x end metavar modus ponens not0 metavar var y end metavar = metavar var x end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar cut 1rule repetition modus ponens not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar conclude not0 not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var x end metavar = metavar var y end metavar imply not0 metavar var y end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma negativeLessPositive as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 - metavar var x end metavar <= metavar var x end metavar imply not0 not0 - metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma negativeLessPositive as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer prop lemma first conjunct modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut lemma leqAddition modus ponens 0 <= metavar var x end metavar conclude 0 + - metavar var x end metavar <= metavar var x end metavar + - metavar var x end metavar cut lemma plus0Left conclude 0 + - metavar var x end metavar = - metavar var x end metavar cut axiom negative conclude metavar var x end metavar + - metavar var x end metavar = 0 cut lemma subLeqLeft modus ponens 0 + - metavar var x end metavar = - metavar var x end metavar modus ponens 0 + - metavar var x end metavar <= metavar var x end metavar + - metavar var x end metavar conclude - metavar var x end metavar <= metavar var x end metavar + - metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar + - metavar var x end metavar = 0 modus ponens - metavar var x end metavar <= metavar var x end metavar + - metavar var x end metavar conclude - metavar var x end metavar <= 0 cut lemma leqLessTransitivity modus ponens - metavar var x end metavar <= 0 modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 - metavar var x end metavar <= metavar var x end metavar imply not0 not0 - metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma lessNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 - metavar var y end metavar <= - metavar var x end metavar imply not0 not0 - metavar var y end metavar = - metavar var x end metavar end define end math ] "


" [ math define proof of lemma lessNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer lemma lessLeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma leqNegated modus ponens metavar var x end metavar <= metavar var y end metavar conclude - metavar var y end metavar <= - metavar var x end metavar cut lemma lessNeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar = metavar var y end metavar cut lemma neqNegated modus ponens not0 metavar var x end metavar = metavar var y end metavar conclude not0 - metavar var x end metavar = - metavar var y end metavar cut lemma neqSymmetry modus ponens not0 - metavar var x end metavar = - metavar var y end metavar conclude not0 - metavar var y end metavar = - metavar var x end metavar cut lemma leqNeqLess modus ponens - metavar var y end metavar <= - metavar var x end metavar modus ponens not0 - metavar var y end metavar = - metavar var x end metavar conclude not0 - metavar var y end metavar <= - metavar var x end metavar imply not0 not0 - metavar var y end metavar = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma -0=0 as system Q infer - 0 = 0 end define end math ] "

" [ math define proof of lemma -0=0 as lambda var c dot lambda var x dot proof expand quote system Q infer axiom negative conclude 0 + - 0 = 0 cut axiom plus0 conclude 0 + 0 = 0 cut lemma uniqueNegative modus ponens 0 + - 0 = 0 modus ponens 0 + 0 = 0 conclude - 0 = 0 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma positiveNegated as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 - metavar var x end metavar <= 0 imply not0 not0 - metavar var x end metavar = 0 end define end math ] "

" [ math define proof of lemma positiveNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma lessNegated modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 - metavar var x end metavar <= - 0 imply not0 not0 - metavar var x end metavar = - 0 cut lemma -0=0 conclude - 0 = 0 cut lemma subLessRight modus ponens - 0 = 0 modus ponens not0 - metavar var x end metavar <= - 0 imply not0 not0 - metavar var x end metavar = - 0 conclude not0 - metavar var x end metavar <= 0 imply not0 not0 - metavar var x end metavar = 0 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma nonpositiveNegated as system Q infer all metavar var x end metavar indeed metavar var x end metavar <= 0 infer 0 <= - metavar var x end metavar end define end math ] "

" [ math define proof of lemma nonpositiveNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed metavar var x end metavar <= 0 infer lemma leqNegated modus ponens metavar var x end metavar <= 0 conclude - 0 <= - metavar var x end metavar cut lemma -0=0 conclude - 0 = 0 cut lemma subLeqLeft modus ponens - 0 = 0 modus ponens - 0 <= - metavar var x end metavar conclude 0 <= - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma negativeNegated as system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar end define end math ] "

" [ math define proof of lemma negativeNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma lessNegated modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 - 0 <= - metavar var x end metavar imply not0 not0 - 0 = - metavar var x end metavar cut lemma -0=0 conclude - 0 = 0 cut lemma subLessLeft modus ponens - 0 = 0 modus ponens not0 - 0 <= - metavar var x end metavar imply not0 not0 - 0 = - metavar var x end metavar conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma nonnegativeNegated as system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer - metavar var x end metavar <= 0 end define end math ] "

" [ math define proof of lemma nonnegativeNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer lemma leqNegated modus ponens 0 <= metavar var x end metavar conclude - metavar var x end metavar <= - 0 cut lemma -0=0 conclude - 0 = 0 cut lemma subLeqRight modus ponens - 0 = 0 modus ponens - metavar var x end metavar <= - 0 conclude - metavar var x end metavar <= 0 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma 0<2 as system Q infer not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 end define end math ] "

" [ math define proof of lemma 0<2 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma lessAddition modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 0 + 1 <= 1 + 1 imply not0 not0 0 + 1 = 1 + 1 cut lemma plus0Left conclude 0 + 1 = 1 cut lemma subLessLeft modus ponens 0 + 1 = 1 modus ponens not0 0 + 1 <= 1 + 1 imply not0 not0 0 + 1 = 1 + 1 conclude not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 cut lemma lessTransitivity modus ponens not0 0 <= 1 imply not0 not0 0 = 1 modus ponens not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 conclude not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma 0<1/2 as system Q infer not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 end define end math ] "

" [ math define proof of lemma 0<1/2 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma 0<2 conclude not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 cut prop lemma first conjunct modus ponens not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 conclude 0 <= 1 + 1 cut prop lemma second conjunct modus ponens not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 conclude not0 0 = 1 + 1 cut lemma neqSymmetry modus ponens not0 0 = 1 + 1 conclude not0 1 + 1 = 0 cut lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma x*0=0 conclude 1 + 1 * 0 = 0 cut lemma x*y=zBackwards modus ponens 1 + 1 * 0 = 0 conclude 0 = 0 * 1 + 1 cut lemma subLessLeft modus ponens 0 = 0 * 1 + 1 modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 0 * 1 + 1 <= 1 imply not0 not0 0 * 1 + 1 = 1 cut lemma reciprocal modus ponens not0 1 + 1 = 0 conclude 1 + 1 * 1/ 1 + 1 = 1 cut lemma x*y=zBackwards modus ponens 1 + 1 * 1/ 1 + 1 = 1 conclude 1 = 1/ 1 + 1 * 1 + 1 cut lemma subLessRight modus ponens 1 = 1/ 1 + 1 * 1 + 1 modus ponens not0 0 * 1 + 1 <= 1 imply not0 not0 0 * 1 + 1 = 1 conclude not0 0 * 1 + 1 <= 1/ 1 + 1 * 1 + 1 imply not0 not0 0 * 1 + 1 = 1/ 1 + 1 * 1 + 1 cut lemma lessDivision modus ponens 0 <= 1 + 1 modus ponens not0 0 * 1 + 1 <= 1/ 1 + 1 * 1 + 1 imply not0 not0 0 * 1 + 1 = 1/ 1 + 1 * 1 + 1 conclude not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma positiveHalved as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= 1/ 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var x end metavar end define end math ] "

" [ math define proof of lemma positiveHalved as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma 0<1/2 conclude not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 cut lemma lessMultiplicationLeft modus ponens not0 0 <= 1/ 1 + 1 imply not0 not0 0 = 1/ 1 + 1 modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 1/ 1 + 1 * 0 <= 1/ 1 + 1 * metavar var x end metavar imply not0 not0 1/ 1 + 1 * 0 = 1/ 1 + 1 * metavar var x end metavar cut lemma x*0=0 conclude 1/ 1 + 1 * 0 = 0 cut lemma subLessLeft modus ponens 1/ 1 + 1 * 0 = 0 modus ponens not0 1/ 1 + 1 * 0 <= 1/ 1 + 1 * metavar var x end metavar imply not0 not0 1/ 1 + 1 * 0 = 1/ 1 + 1 * metavar var x end metavar conclude not0 0 <= 1/ 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma fromNot<< as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma fromNot<< as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed prop lemma auto imply conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer prop lemma mt modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma nonnegativeNumerical as system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer | metavar var x end metavar | = metavar var x end metavar end define end math ] "

" [ math define proof of lemma nonnegativeNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer axiom numerical conclude not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = metavar var x end metavar imply not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = - metavar var x end metavar cut prop lemma add double neg modus ponens 0 <= metavar var x end metavar conclude not0 not0 0 <= metavar var x end metavar cut prop lemma to negated and(1) modus ponens not0 not0 0 <= metavar var x end metavar conclude not0 not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = - metavar var x end metavar cut prop lemma negate second disjunct modus ponens not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = metavar var x end metavar imply not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = - metavar var x end metavar modus ponens not0 not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = - metavar var x end metavar conclude not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = metavar var x end metavar cut prop lemma second conjunct modus ponens not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveNumerical as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer | metavar var x end metavar | = metavar var x end metavar end define end math ] "

" [ math define proof of lemma positiveNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma lessLeq modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma negativeNumerical as system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer | metavar var x end metavar | = - metavar var x end metavar end define end math ] "

" [ math define proof of lemma negativeNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer axiom numerical conclude not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = metavar var x end metavar imply not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = - metavar var x end metavar cut lemma fromLess modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 0 <= metavar var x end metavar cut prop lemma to negated and(1) modus ponens not0 0 <= metavar var x end metavar conclude not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = metavar var x end metavar cut prop lemma negate first disjunct modus ponens not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = metavar var x end metavar imply not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = - metavar var x end metavar modus ponens not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = metavar var x end metavar conclude not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = - metavar var x end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | = - metavar var x end metavar conclude | metavar var x end metavar | = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma nonpositiveNumerical as system Q infer all metavar var x end metavar indeed metavar var x end metavar <= 0 infer | metavar var x end metavar | = - metavar var x end metavar end define end math ] "

" [ math define proof of lemma nonpositiveNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma negativeNumerical modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude | metavar var x end metavar | = - metavar var x end metavar cut all metavar var x end metavar indeed metavar var x end metavar = 0 infer lemma eqSymmetry modus ponens metavar var x end metavar = 0 conclude 0 = metavar var x end metavar cut lemma eqLeq modus ponens 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma -0=0 conclude - 0 = 0 cut lemma eqSymmetry modus ponens - 0 = 0 conclude 0 = - 0 cut lemma eqNegated modus ponens 0 = metavar var x end metavar conclude - 0 = - metavar var x end metavar cut lemma eqTransitivity5 modus ponens | metavar var x end metavar | = metavar var x end metavar modus ponens metavar var x end metavar = 0 modus ponens 0 = - 0 modus ponens - 0 = - metavar var x end metavar conclude | metavar var x end metavar | = - metavar var x end metavar cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer | metavar var x end metavar | = - metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply | metavar var x end metavar | = - metavar var x end metavar cut 1rule deduction modus ponens all metavar var x end metavar indeed metavar var x end metavar = 0 infer | metavar var x end metavar | = - metavar var x end metavar conclude metavar var x end metavar = 0 imply | metavar var x end metavar | = - metavar var x end metavar cut metavar var x end metavar <= 0 infer lemma leqLessEq modus ponens metavar var x end metavar <= 0 conclude not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply metavar var x end metavar = 0 cut prop lemma from disjuncts modus ponens not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply metavar var x end metavar = 0 modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply | metavar var x end metavar | = - metavar var x end metavar modus ponens metavar var x end metavar = 0 imply | metavar var x end metavar | = - metavar var x end metavar conclude | metavar var x end metavar | = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma |0|=0 as system Q infer | 0 | = 0 end define end math ] "

" [ math define proof of lemma |0|=0 as lambda var c dot lambda var x dot proof expand quote system Q infer axiom leqReflexivity conclude 0 <= 0 cut lemma nonnegativeNumerical modus ponens 0 <= 0 conclude | 0 | = 0 end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma 0<=|x| as system Q infer all metavar var x end metavar indeed 0 <= | metavar var x end metavar | end define end math ] "

" [ math define proof of lemma 0<=|x| as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma eqSymmetry modus ponens | metavar var x end metavar | = metavar var x end metavar conclude metavar var x end metavar = | metavar var x end metavar | cut lemma subLeqRight modus ponens metavar var x end metavar = | metavar var x end metavar | modus ponens 0 <= metavar var x end metavar conclude 0 <= | metavar var x end metavar | cut all metavar var x end metavar indeed not0 0 <= metavar var x end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude | metavar var x end metavar | = - metavar var x end metavar cut lemma eqSymmetry modus ponens | metavar var x end metavar | = - metavar var x end metavar conclude - metavar var x end metavar = | metavar var x end metavar | cut lemma negativeNegated modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar cut lemma lessLeq modus ponens not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar conclude 0 <= - metavar var x end metavar cut lemma subLeqRight modus ponens - metavar var x end metavar = | metavar var x end metavar | modus ponens 0 <= - metavar var x end metavar conclude 0 <= | metavar var x end metavar | cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed 0 <= metavar var x end metavar infer 0 <= | metavar var x end metavar | conclude 0 <= metavar var x end metavar imply 0 <= | metavar var x end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed not0 0 <= metavar var x end metavar infer 0 <= | metavar var x end metavar | conclude not0 0 <= metavar var x end metavar imply 0 <= | metavar var x end metavar | cut prop lemma from negations modus ponens 0 <= metavar var x end metavar imply 0 <= | metavar var x end metavar | modus ponens not0 0 <= metavar var x end metavar imply 0 <= | metavar var x end metavar | conclude 0 <= | metavar var x end metavar | end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma sameNumerical as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer | metavar var x end metavar | = | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma sameNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar = metavar var y end metavar modus ponens 0 <= metavar var x end metavar conclude 0 <= metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var y end metavar conclude | metavar var y end metavar | = metavar var y end metavar cut lemma eqSymmetry modus ponens | metavar var y end metavar | = metavar var y end metavar conclude metavar var y end metavar = | metavar var y end metavar | cut lemma eqTransitivity4 modus ponens | metavar var x end metavar | = metavar var x end metavar modus ponens metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar = | metavar var y end metavar | conclude | metavar var x end metavar | = | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude | metavar var x end metavar | = - metavar var x end metavar cut lemma subLessLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar = 0 conclude | metavar var y end metavar | = - metavar var y end metavar cut lemma eqSymmetry modus ponens | metavar var y end metavar | = - metavar var y end metavar conclude - metavar var y end metavar = | metavar var y end metavar | cut lemma eqNegated modus ponens metavar var x end metavar = metavar var y end metavar conclude - metavar var x end metavar = - metavar var y end metavar cut lemma eqTransitivity4 modus ponens | metavar var x end metavar | = - metavar var x end metavar modus ponens - metavar var x end metavar = - metavar var y end metavar modus ponens - metavar var y end metavar = | metavar var y end metavar | conclude | metavar var x end metavar | = | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer | metavar var x end metavar | = | metavar var y end metavar | conclude 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply | metavar var x end metavar | = | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer | metavar var x end metavar | = | metavar var y end metavar | conclude not0 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply | metavar var x end metavar | = | metavar var y end metavar | cut prop lemma from negations modus ponens 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply | metavar var x end metavar | = | metavar var y end metavar | modus ponens not0 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply | metavar var x end metavar | = | metavar var y end metavar | conclude metavar var x end metavar = metavar var y end metavar imply | metavar var x end metavar | = | metavar var y end metavar | cut 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply | metavar var x end metavar | = | metavar var y end metavar | modus ponens metavar var x end metavar = metavar var y end metavar conclude | metavar var x end metavar | = | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameSeries(Gen) as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var sz end metavar indeed metavar var m end metavar in0 N infer metavar var n end metavar in0 N infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sz end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sz end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar infer metavar var m end metavar = metavar var n end metavar infer metavar var fx end metavar = metavar var fy end metavar infer [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var n end metavar ] end define end math ] "



" [ math define proof of lemma sameSeries(Gen) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var sz end metavar indeed metavar var m end metavar in0 N infer metavar var n end metavar in0 N infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sz end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sz end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar infer metavar var m end metavar = metavar var n end metavar infer metavar var fx end metavar = metavar var fy end metavar infer lemma memberOfSeries modus ponens metavar var m end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sz end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair in0 metavar var fx end metavar cut lemma set equality nec condition(1) modus ponens metavar var fx end metavar = metavar var fy end metavar modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair in0 metavar var fx end metavar conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair in0 metavar var fy end metavar cut lemma memberOfSeries modus ponens metavar var n end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sz end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar conclude zermelo pair zermelo pair metavar var n end metavar comma metavar var n end metavar end pair comma zermelo pair metavar var n end metavar comma [ metavar var fy end metavar ; metavar var n end metavar ] end pair end pair in0 metavar var fy end metavar cut lemma uniqueMember modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sz end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair in0 metavar var fy end metavar modus ponens zermelo pair zermelo pair metavar var n end metavar comma metavar var n end metavar end pair comma zermelo pair metavar var n end metavar comma [ metavar var fy end metavar ; metavar var n end metavar ] end pair end pair in0 metavar var fy end metavar modus ponens metavar var m end metavar = metavar var n end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var n end metavar ] end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma equalsSameF as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar = metavar var fy end metavar infer for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end define end math ] "

" [ math define proof of lemma equalsSameF as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar = metavar var fy end metavar infer not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var infer 0 <= object var var m end var infer axiom natType conclude object var var m end var in0 N cut axiom seriesType conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut axiom seriesType conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar cut lemma eqReflexivity conclude object var var m end var = object var var m end var cut lemma sameSeries(Gen) modus ponens object var var m end var in0 N modus ponens object var var m end var in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar modus ponens object var var m end var = object var var m end var modus ponens metavar var fx end metavar = metavar var fy end metavar conclude [ metavar var fx end metavar ; object var var m end var ] = [ metavar var fy end metavar ; object var var m end var ] cut lemma positiveToLeft(Eq)(1 term) modus ponens [ metavar var fx end metavar ; object var var m end var ] = [ metavar var fy end metavar ; object var var m end var ] conclude [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] = 0 cut lemma sameNumerical modus ponens [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] = 0 conclude | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = | 0 | cut lemma |0|=0 conclude | 0 | = 0 cut lemma eqTransitivity modus ponens | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = | 0 | modus ponens | 0 | = 0 conclude | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 0 cut lemma eqSymmetry modus ponens | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 0 conclude 0 = | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | cut lemma subLessLeft modus ponens 0 = | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | modus ponens not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var conclude not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed metavar var fx end metavar = metavar var fy end metavar infer not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var infer 0 <= object var var m end var infer not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude metavar var fx end metavar = metavar var fy end metavar imply not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply 0 <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut metavar var fx end metavar = metavar var fy end metavar infer 1rule mp modus ponens metavar var fx end metavar = metavar var fy end metavar imply not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply 0 <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var modus ponens metavar var fx end metavar = metavar var fy end metavar conclude not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply 0 <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule gen modus ponens not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply 0 <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply 0 <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut pred lemma intro exist at 0 modus ponens for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply 0 <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule gen modus ponens not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule repetition modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end quote state proof state cache var c end expand end define end math ] "







" [ math define statement of lemma signNumerical(+) as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer | metavar var x end metavar | = | - metavar var x end metavar | end define end math ] "

" [ math define proof of lemma signNumerical(+) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma positiveNumerical modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma positiveNegated modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 - metavar var x end metavar <= 0 imply not0 not0 - metavar var x end metavar = 0 cut lemma negativeNumerical modus ponens not0 - metavar var x end metavar <= 0 imply not0 not0 - metavar var x end metavar = 0 conclude | - metavar var x end metavar | = - - metavar var x end metavar cut lemma doubleMinus conclude - - metavar var x end metavar = metavar var x end metavar cut lemma eqTransitivity modus ponens | - metavar var x end metavar | = - - metavar var x end metavar modus ponens - - metavar var x end metavar = metavar var x end metavar conclude | - metavar var x end metavar | = metavar var x end metavar cut lemma eqSymmetry modus ponens | - metavar var x end metavar | = metavar var x end metavar conclude metavar var x end metavar = | - metavar var x end metavar | cut lemma eqTransitivity modus ponens | metavar var x end metavar | = metavar var x end metavar modus ponens metavar var x end metavar = | - metavar var x end metavar | conclude | metavar var x end metavar | = | - metavar var x end metavar | end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma signNumerical as system Q infer all metavar var x end metavar indeed | metavar var x end metavar | = | - metavar var x end metavar | end define end math ] "

" [ math define proof of lemma signNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma negativeNegated modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar cut lemma signNumerical(+) modus ponens not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar conclude | - metavar var x end metavar | = | - - metavar var x end metavar | cut lemma doubleMinus conclude - - metavar var x end metavar = metavar var x end metavar cut lemma sameNumerical modus ponens - - metavar var x end metavar = metavar var x end metavar conclude | - - metavar var x end metavar | = | metavar var x end metavar | cut lemma eqTransitivity modus ponens | - metavar var x end metavar | = | - - metavar var x end metavar | modus ponens | - - metavar var x end metavar | = | metavar var x end metavar | conclude | - metavar var x end metavar | = | metavar var x end metavar | cut lemma eqSymmetry modus ponens | - metavar var x end metavar | = | metavar var x end metavar | conclude | metavar var x end metavar | = | - metavar var x end metavar | cut all metavar var x end metavar indeed metavar var x end metavar = 0 infer lemma eqNegated modus ponens metavar var x end metavar = 0 conclude - metavar var x end metavar = - 0 cut lemma -0=0 conclude - 0 = 0 cut lemma eqSymmetry modus ponens metavar var x end metavar = 0 conclude 0 = metavar var x end metavar cut lemma eqTransitivity4 modus ponens - metavar var x end metavar = - 0 modus ponens - 0 = 0 modus ponens 0 = metavar var x end metavar conclude - metavar var x end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens - metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar = - metavar var x end metavar cut lemma sameNumerical modus ponens metavar var x end metavar = - metavar var x end metavar conclude | metavar var x end metavar | = | - metavar var x end metavar | cut all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma signNumerical(+) modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude | metavar var x end metavar | = | - metavar var x end metavar | cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer | metavar var x end metavar | = | - metavar var x end metavar | conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply | metavar var x end metavar | = | - metavar var x end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed metavar var x end metavar = 0 infer | metavar var x end metavar | = | - metavar var x end metavar | conclude metavar var x end metavar = 0 imply | metavar var x end metavar | = | - metavar var x end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer | metavar var x end metavar | = | - metavar var x end metavar | conclude not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar imply | metavar var x end metavar | = | - metavar var x end metavar | cut lemma lessTotality conclude not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply not0 metavar var x end metavar = 0 imply not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar cut prop lemma from three disjuncts modus ponens not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply not0 metavar var x end metavar = 0 imply not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply | metavar var x end metavar | = | - metavar var x end metavar | modus ponens metavar var x end metavar = 0 imply | metavar var x end metavar | = | - metavar var x end metavar | modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar imply | metavar var x end metavar | = | - metavar var x end metavar | conclude | metavar var x end metavar | = | - metavar var x end metavar | end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma times(-1) as system Q infer all metavar var x end metavar indeed metavar var x end metavar * - 1 = - metavar var x end metavar end define end math ] "

" [ math define proof of lemma times(-1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom negative conclude 1 + - 1 = 0 cut axiom plusCommutativity conclude - 1 + 1 = 1 + - 1 cut lemma eqTransitivity modus ponens - 1 + 1 = 1 + - 1 modus ponens 1 + - 1 = 0 conclude - 1 + 1 = 0 cut lemma eqMultiplicationLeft modus ponens - 1 + 1 = 0 conclude metavar var x end metavar * - 1 + 1 = metavar var x end metavar * 0 cut lemma x*0=0 conclude metavar var x end metavar * 0 = 0 cut lemma eqTransitivity modus ponens metavar var x end metavar * - 1 + 1 = metavar var x end metavar * 0 modus ponens metavar var x end metavar * 0 = 0 conclude metavar var x end metavar * - 1 + 1 = 0 cut axiom distribution conclude metavar var x end metavar * - 1 + 1 = metavar var x end metavar * - 1 + metavar var x end metavar * 1 cut lemma eqSymmetry modus ponens metavar var x end metavar * - 1 + 1 = metavar var x end metavar * - 1 + metavar var x end metavar * 1 conclude metavar var x end metavar * - 1 + metavar var x end metavar * 1 = metavar var x end metavar * - 1 + 1 cut lemma eqTransitivity modus ponens metavar var x end metavar * - 1 + metavar var x end metavar * 1 = metavar var x end metavar * - 1 + 1 modus ponens metavar var x end metavar * - 1 + 1 = 0 conclude metavar var x end metavar * - 1 + metavar var x end metavar * 1 = 0 cut lemma positiveToRight(Eq) modus ponens metavar var x end metavar * - 1 + metavar var x end metavar * 1 = 0 conclude metavar var x end metavar * - 1 = 0 + - metavar var x end metavar * 1 cut lemma plus0Left conclude 0 + - metavar var x end metavar * 1 = - metavar var x end metavar * 1 cut lemma eqTransitivity modus ponens metavar var x end metavar * - 1 = 0 + - metavar var x end metavar * 1 modus ponens 0 + - metavar var x end metavar * 1 = - metavar var x end metavar * 1 conclude metavar var x end metavar * - 1 = - metavar var x end metavar * 1 cut axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut lemma eqNegated modus ponens metavar var x end metavar * 1 = metavar var x end metavar conclude - metavar var x end metavar * 1 = - metavar var x end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar * - 1 = - metavar var x end metavar * 1 modus ponens - metavar var x end metavar * 1 = - metavar var x end metavar conclude metavar var x end metavar * - 1 = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma times(-1)Left as system Q infer all metavar var x end metavar indeed - 1 * metavar var x end metavar = - metavar var x end metavar end define end math ] "

" [ math define proof of lemma times(-1)Left as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma times(-1) conclude metavar var x end metavar * - 1 = - metavar var x end metavar cut axiom timesCommutativity conclude - 1 * metavar var x end metavar = metavar var x end metavar * - 1 cut lemma eqTransitivity modus ponens - 1 * metavar var x end metavar = metavar var x end metavar * - 1 modus ponens metavar var x end metavar * - 1 = - metavar var x end metavar conclude - 1 * metavar var x end metavar = - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma -x-y=-(x+y) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma -x-y=-(x+y) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma times(-1)Left conclude - 1 * metavar var x end metavar = - metavar var x end metavar cut lemma times(-1)Left conclude - 1 * metavar var y end metavar = - metavar var y end metavar cut lemma addEquations modus ponens - 1 * metavar var x end metavar = - metavar var x end metavar modus ponens - 1 * metavar var y end metavar = - metavar var y end metavar conclude - 1 * metavar var x end metavar + - 1 * metavar var y end metavar = - metavar var x end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens - 1 * metavar var x end metavar + - 1 * metavar var y end metavar = - metavar var x end metavar + - metavar var y end metavar conclude - metavar var x end metavar + - metavar var y end metavar = - 1 * metavar var x end metavar + - 1 * metavar var y end metavar cut lemma distributionOut conclude - 1 * metavar var x end metavar + - 1 * metavar var y end metavar = - 1 * metavar var x end metavar + metavar var y end metavar cut lemma times(-1)Left conclude - 1 * metavar var x end metavar + metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar cut lemma eqTransitivity4 modus ponens - metavar var x end metavar + - metavar var y end metavar = - 1 * metavar var x end metavar + - 1 * metavar var y end metavar modus ponens - 1 * metavar var x end metavar + - 1 * metavar var y end metavar = - 1 * metavar var x end metavar + metavar var y end metavar modus ponens - 1 * metavar var x end metavar + metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma minusNegated as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar end define end math ] "

" [ math define proof of lemma minusNegated as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma eqAddition modus ponens - - metavar var y end metavar = metavar var y end metavar conclude - - metavar var y end metavar + - metavar var x end metavar = metavar var y end metavar + - metavar var x end metavar cut lemma eqSymmetry modus ponens - - metavar var y end metavar + - metavar var x end metavar = metavar var y end metavar + - metavar var x end metavar conclude metavar var y end metavar + - metavar var x end metavar = - - metavar var y end metavar + - metavar var x end metavar cut lemma -x-y=-(x+y) conclude - - metavar var y end metavar + - metavar var x end metavar = - - metavar var y end metavar + metavar var x end metavar cut axiom plusCommutativity conclude - metavar var y end metavar + metavar var x end metavar = metavar var x end metavar + - metavar var y end metavar cut lemma eqNegated modus ponens - metavar var y end metavar + metavar var x end metavar = metavar var x end metavar + - metavar var y end metavar conclude - - metavar var y end metavar + metavar var x end metavar = - metavar var x end metavar + - metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var y end metavar + - metavar var x end metavar = - - metavar var y end metavar + - metavar var x end metavar modus ponens - - metavar var y end metavar + - metavar var x end metavar = - - metavar var y end metavar + metavar var x end metavar modus ponens - - metavar var y end metavar + metavar var x end metavar = - metavar var x end metavar + - metavar var y end metavar conclude metavar var y end metavar + - metavar var x end metavar = - metavar var x end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar + - metavar var x end metavar = - metavar var x end metavar + - metavar var y end metavar conclude - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma numericalDifference as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed | metavar var x end metavar + - metavar var y end metavar | = | metavar var y end metavar + - metavar var x end metavar | end define end math ] "

" [ math define proof of lemma numericalDifference as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma signNumerical conclude | metavar var x end metavar + - metavar var y end metavar | = | - metavar var x end metavar + - metavar var y end metavar | cut lemma minusNegated conclude - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar conclude | - metavar var x end metavar + - metavar var y end metavar | = | metavar var y end metavar + - metavar var x end metavar | cut lemma eqTransitivity modus ponens | metavar var x end metavar + - metavar var y end metavar | = | - metavar var x end metavar + - metavar var y end metavar | modus ponens | - metavar var x end metavar + - metavar var y end metavar | = | metavar var y end metavar + - metavar var x end metavar | conclude | metavar var x end metavar + - metavar var y end metavar | = | metavar var y end metavar + - metavar var x end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma splitNumericalSumHelper as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed | - metavar var x end metavar + - metavar var y end metavar | <= | - metavar var x end metavar | + | - metavar var y end metavar | infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalSumHelper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed | - metavar var x end metavar + - metavar var y end metavar | <= | - metavar var x end metavar | + | - metavar var y end metavar | infer lemma signNumerical conclude | metavar var x end metavar | = | - metavar var x end metavar | cut lemma signNumerical conclude | metavar var y end metavar | = | - metavar var y end metavar | cut lemma addEquations modus ponens | metavar var x end metavar | = | - metavar var x end metavar | modus ponens | metavar var y end metavar | = | - metavar var y end metavar | conclude | metavar var x end metavar | + | metavar var y end metavar | = | - metavar var x end metavar | + | - metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar | + | metavar var y end metavar | = | - metavar var x end metavar | + | - metavar var y end metavar | conclude | - metavar var x end metavar | + | - metavar var y end metavar | = | metavar var x end metavar | + | metavar var y end metavar | cut lemma -x-y=-(x+y) conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar conclude | - metavar var x end metavar + - metavar var y end metavar | = | - metavar var x end metavar + metavar var y end metavar | cut lemma signNumerical conclude | metavar var x end metavar + metavar var y end metavar | = | - metavar var x end metavar + metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar + metavar var y end metavar | = | - metavar var x end metavar + metavar var y end metavar | conclude | - metavar var x end metavar + metavar var y end metavar | = | metavar var x end metavar + metavar var y end metavar | cut lemma eqTransitivity modus ponens | - metavar var x end metavar + - metavar var y end metavar | = | - metavar var x end metavar + metavar var y end metavar | modus ponens | - metavar var x end metavar + metavar var y end metavar | = | metavar var x end metavar + metavar var y end metavar | conclude | - metavar var x end metavar + - metavar var y end metavar | = | metavar var x end metavar + metavar var y end metavar | cut lemma subLeqRight modus ponens | - metavar var x end metavar | + | - metavar var y end metavar | = | metavar var x end metavar | + | metavar var y end metavar | modus ponens | - metavar var x end metavar + - metavar var y end metavar | <= | - metavar var x end metavar | + | - metavar var y end metavar | conclude | - metavar var x end metavar + - metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma subLeqLeft modus ponens | - metavar var x end metavar + - metavar var y end metavar | = | metavar var x end metavar + metavar var y end metavar | modus ponens | - metavar var x end metavar + - metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma splitNumericalSum(++) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalSum(++) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma addEquations(Leq) modus ponens 0 <= metavar var x end metavar modus ponens 0 <= metavar var y end metavar conclude 0 + 0 <= metavar var x end metavar + metavar var y end metavar cut axiom plus0 conclude 0 + 0 = 0 cut lemma subLeqLeft modus ponens 0 + 0 = 0 modus ponens 0 + 0 <= metavar var x end metavar + metavar var y end metavar conclude 0 <= metavar var x end metavar + metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar + metavar var y end metavar conclude | metavar var x end metavar + metavar var y end metavar | = metavar var x end metavar + metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var y end metavar conclude | metavar var y end metavar | = metavar var y end metavar cut lemma addEquations modus ponens | metavar var x end metavar | = metavar var x end metavar modus ponens | metavar var y end metavar | = metavar var y end metavar conclude | metavar var x end metavar | + | metavar var y end metavar | = metavar var x end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens | metavar var x end metavar | + | metavar var y end metavar | = metavar var x end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar = | metavar var x end metavar | + | metavar var y end metavar | cut lemma eqTransitivity modus ponens | metavar var x end metavar + metavar var y end metavar | = metavar var x end metavar + metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar = | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | = | metavar var x end metavar | + | metavar var y end metavar | cut lemma eqLeq modus ponens | metavar var x end metavar + metavar var y end metavar | = | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma splitNumericalSum(--) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalSum(--) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer lemma nonpositiveNegated modus ponens metavar var x end metavar <= 0 conclude 0 <= - metavar var x end metavar cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma splitNumericalSum(++) modus ponens 0 <= - metavar var x end metavar modus ponens 0 <= - metavar var y end metavar conclude | - metavar var x end metavar + - metavar var y end metavar | <= | - metavar var x end metavar | + | - metavar var y end metavar | cut lemma splitNumericalSumHelper modus ponens | - metavar var x end metavar + - metavar var y end metavar | <= | - metavar var x end metavar | + | - metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma splitNumericalSum(+-, smallNegative) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var y end metavar | <= | metavar var x end metavar | infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalSum(+-, smallNegative) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var y end metavar | <= | metavar var x end metavar | infer lemma leqAdditionLeft modus ponens metavar var y end metavar <= 0 conclude metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar + 0 cut axiom plus0 conclude metavar var x end metavar + 0 = metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar + 0 = metavar var x end metavar modus ponens metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar + 0 conclude metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar cut lemma positiveToRight(Leq)(1 term) modus ponens | metavar var y end metavar | <= | metavar var x end metavar | conclude 0 <= | metavar var x end metavar | + - | metavar var y end metavar | cut lemma nonpositiveNumerical modus ponens metavar var y end metavar <= 0 conclude | metavar var y end metavar | = - metavar var y end metavar cut lemma eqNegated modus ponens | metavar var y end metavar | = - metavar var y end metavar conclude - | metavar var y end metavar | = - - metavar var y end metavar cut lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma eqTransitivity modus ponens - | metavar var y end metavar | = - - metavar var y end metavar modus ponens - - metavar var y end metavar = metavar var y end metavar conclude - | metavar var y end metavar | = metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma addEquations modus ponens | metavar var x end metavar | = metavar var x end metavar modus ponens - | metavar var y end metavar | = metavar var y end metavar conclude | metavar var x end metavar | + - | metavar var y end metavar | = metavar var x end metavar + metavar var y end metavar cut lemma subLeqRight modus ponens | metavar var x end metavar | + - | metavar var y end metavar | = metavar var x end metavar + metavar var y end metavar modus ponens 0 <= | metavar var x end metavar | + - | metavar var y end metavar | conclude 0 <= metavar var x end metavar + metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar + metavar var y end metavar conclude | metavar var x end metavar + metavar var y end metavar | = metavar var x end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens | metavar var x end metavar + metavar var y end metavar | = metavar var x end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar = | metavar var x end metavar + metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar | = metavar var x end metavar conclude metavar var x end metavar = | metavar var x end metavar | cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var y end metavar = | metavar var x end metavar + metavar var y end metavar | modus ponens metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar conclude | metavar var x end metavar + metavar var y end metavar | <= metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar = | metavar var x end metavar | modus ponens | metavar var x end metavar + metavar var y end metavar | <= metavar var x end metavar conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma splitNumericalSum(+-, bigNegative) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer not0 | metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | metavar var x end metavar | = | metavar var y end metavar | infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalSum(+-, bigNegative) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer not0 | metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | metavar var x end metavar | = | metavar var y end metavar | infer lemma nonnegativeNegated modus ponens 0 <= metavar var x end metavar conclude - metavar var x end metavar <= 0 cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma signNumerical conclude | metavar var x end metavar | = | - metavar var x end metavar | cut lemma subLessLeft modus ponens | metavar var x end metavar | = | - metavar var x end metavar | modus ponens not0 | metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | metavar var x end metavar | = | metavar var y end metavar | conclude not0 | - metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | - metavar var x end metavar | = | metavar var y end metavar | cut lemma signNumerical conclude | metavar var y end metavar | = | - metavar var y end metavar | cut lemma subLessRight modus ponens | metavar var y end metavar | = | - metavar var y end metavar | modus ponens not0 | - metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | - metavar var x end metavar | = | metavar var y end metavar | conclude not0 | - metavar var x end metavar | <= | - metavar var y end metavar | imply not0 not0 | - metavar var x end metavar | = | - metavar var y end metavar | cut lemma lessLeq modus ponens not0 | - metavar var x end metavar | <= | - metavar var y end metavar | imply not0 not0 | - metavar var x end metavar | = | - metavar var y end metavar | conclude | - metavar var x end metavar | <= | - metavar var y end metavar | cut lemma splitNumericalSum(+-, smallNegative) modus ponens 0 <= - metavar var y end metavar modus ponens - metavar var x end metavar <= 0 modus ponens | - metavar var x end metavar | <= | - metavar var y end metavar | conclude | - metavar var y end metavar + - metavar var x end metavar | <= | - metavar var y end metavar | cut lemma signNumerical conclude | metavar var x end metavar + metavar var y end metavar | = | - metavar var x end metavar + metavar var y end metavar | cut lemma -x-y=-(x+y) conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar cut axiom plusCommutativity conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar cut lemma equality modus ponens - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar modus ponens - metavar var x end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar conclude - metavar var x end metavar + metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar + metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar conclude | - metavar var x end metavar + metavar var y end metavar | = | - metavar var y end metavar + - metavar var x end metavar | cut lemma eqTransitivity modus ponens | metavar var x end metavar + metavar var y end metavar | = | - metavar var x end metavar + metavar var y end metavar | modus ponens | - metavar var x end metavar + metavar var y end metavar | = | - metavar var y end metavar + - metavar var x end metavar | conclude | metavar var x end metavar + metavar var y end metavar | = | - metavar var y end metavar + - metavar var x end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar + metavar var y end metavar | = | - metavar var y end metavar + - metavar var x end metavar | conclude | - metavar var y end metavar + - metavar var x end metavar | = | metavar var x end metavar + metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var y end metavar | = | - metavar var y end metavar | conclude | - metavar var y end metavar | = | metavar var y end metavar | cut lemma subLeqLeft modus ponens | - metavar var y end metavar + - metavar var x end metavar | = | metavar var x end metavar + metavar var y end metavar | modus ponens | - metavar var y end metavar + - metavar var x end metavar | <= | - metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | - metavar var y end metavar | cut lemma subLeqRight modus ponens | - metavar var y end metavar | = | metavar var y end metavar | modus ponens | metavar var x end metavar + metavar var y end metavar | <= | - metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma splitNumericalSum(+-) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalSum(+-) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed | metavar var y end metavar | <= | metavar var x end metavar | infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(+-, smallNegative) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 modus ponens | metavar var y end metavar | <= | metavar var x end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | cut lemma 0<=|x| conclude 0 <= | metavar var y end metavar | cut lemma leqAdditionLeft modus ponens 0 <= | metavar var y end metavar | conclude | metavar var x end metavar | + 0 <= | metavar var x end metavar | + | metavar var y end metavar | cut axiom plus0 conclude | metavar var x end metavar | + 0 = | metavar var x end metavar | cut lemma subLeqLeft modus ponens | metavar var x end metavar | + 0 = | metavar var x end metavar | modus ponens | metavar var x end metavar | + 0 <= | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var x end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma leqTransitivity modus ponens | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | modus ponens | metavar var x end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 | metavar var y end metavar | <= | metavar var x end metavar | infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma toLess modus ponens not0 | metavar var y end metavar | <= | metavar var x end metavar | conclude not0 | metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | metavar var x end metavar | = | metavar var y end metavar | cut lemma splitNumericalSum(+-, bigNegative) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 modus ponens not0 | metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | metavar var x end metavar | = | metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var y end metavar | cut lemma 0<=|x| conclude 0 <= | metavar var x end metavar | cut lemma leqAddition modus ponens 0 <= | metavar var x end metavar | conclude 0 + | metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma plus0Left conclude 0 + | metavar var y end metavar | = | metavar var y end metavar | cut lemma subLeqLeft modus ponens 0 + | metavar var y end metavar | = | metavar var y end metavar | modus ponens 0 + | metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma leqTransitivity modus ponens | metavar var x end metavar + metavar var y end metavar | <= | metavar var y end metavar | modus ponens | metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed | metavar var y end metavar | <= | metavar var x end metavar | infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var y end metavar | <= | metavar var x end metavar | imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 | metavar var y end metavar | <= | metavar var x end metavar | infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude not0 | metavar var y end metavar | <= | metavar var x end metavar | imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer prop lemma from negations modus ponens | metavar var y end metavar | <= | metavar var x end metavar | imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | modus ponens not0 | metavar var y end metavar | <= | metavar var x end metavar | imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut prop lemma mp2 modus ponens 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma splitNumericalSum(-+) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalSum(-+) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer lemma nonpositiveNegated modus ponens metavar var x end metavar <= 0 conclude 0 <= - metavar var x end metavar cut lemma nonnegativeNegated modus ponens 0 <= metavar var y end metavar conclude - metavar var y end metavar <= 0 cut lemma splitNumericalSum(+-) modus ponens 0 <= - metavar var x end metavar modus ponens - metavar var y end metavar <= 0 conclude | - metavar var x end metavar + - metavar var y end metavar | <= | - metavar var x end metavar | + | - metavar var y end metavar | cut lemma splitNumericalSumHelper modus ponens | - metavar var x end metavar + - metavar var y end metavar | <= | - metavar var x end metavar | + | - metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma splitNumericalSum as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end define end math ] "



" [ math define proof of lemma splitNumericalSum as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma splitNumericalSum(++) modus ponens 0 <= metavar var x end metavar modus ponens 0 <= metavar var y end metavar conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(+-) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer lemma splitNumericalSum(-+) modus ponens metavar var x end metavar <= 0 modus ponens 0 <= metavar var y end metavar conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(--) modus ponens metavar var x end metavar <= 0 modus ponens metavar var y end metavar <= 0 conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude metavar var x end metavar <= 0 imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | modus ponens metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude 0 <= metavar var y end metavar imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | modus ponens metavar var x end metavar <= 0 imply metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var y end metavar imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | modus ponens metavar var y end metavar <= 0 imply | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma insertMiddleTerm(Sum) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma insertMiddleTerm(Sum) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma three2threeTerms conclude metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var x end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma eqAddition modus ponens metavar var x end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar cut axiom plusAssociativity conclude metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar modus ponens metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma insertMiddleTerm(Numerical) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + metavar var y end metavar | end define end math ] "

" [ math define proof of lemma insertMiddleTerm(Numerical) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma splitNumericalSum conclude | metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + metavar var y end metavar | cut lemma insertMiddleTerm(Sum) conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar cut lemma sameNumerical modus ponens metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar conclude | metavar var x end metavar + metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar + metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar | conclude | metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar | = | metavar var x end metavar + metavar var y end metavar | cut lemma subLeqLeft modus ponens | metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar | = | metavar var x end metavar + metavar var y end metavar | modus ponens | metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

(*** REGNESTYKKER ***)



" [ math define statement of lemma insertMiddleTerm(Difference) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma insertMiddleTerm(Difference) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma insertMiddleTerm(Sum) conclude metavar var x end metavar + - metavar var y end metavar = metavar var x end metavar + - - metavar var z end metavar + - metavar var z end metavar + - metavar var y end metavar cut lemma doubleMinus conclude - - metavar var z end metavar = metavar var z end metavar cut lemma eqAdditionLeft modus ponens - - metavar var z end metavar = metavar var z end metavar conclude metavar var x end metavar + - - metavar var z end metavar = metavar var x end metavar + metavar var z end metavar cut axiom plusCommutativity conclude - metavar var z end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var z end metavar cut lemma -x-y=-(x+y) conclude - metavar var y end metavar + - metavar var z end metavar = - metavar var y end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens - metavar var z end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var z end metavar modus ponens - metavar var y end metavar + - metavar var z end metavar = - metavar var y end metavar + metavar var z end metavar conclude - metavar var z end metavar + - metavar var y end metavar = - metavar var y end metavar + metavar var z end metavar cut lemma addEquations modus ponens metavar var x end metavar + - - metavar var z end metavar = metavar var x end metavar + metavar var z end metavar modus ponens - metavar var z end metavar + - metavar var y end metavar = - metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar + - - metavar var z end metavar + - metavar var z end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var y end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var y end metavar = metavar var x end metavar + - - metavar var z end metavar + - metavar var z end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + - - metavar var z end metavar + - metavar var z end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar + - metavar var y end metavar = metavar var x end metavar + metavar var z end metavar + - metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma distributionOutLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var y end metavar * metavar var x end metavar + metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma distributionOutLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed axiom timesCommutativity conclude metavar var y end metavar * metavar var x end metavar = metavar var x end metavar * metavar var y end metavar cut axiom timesCommutativity conclude metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var z end metavar cut lemma addEquations modus ponens metavar var y end metavar * metavar var x end metavar = metavar var x end metavar * metavar var y end metavar modus ponens metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var z end metavar conclude metavar var y end metavar * metavar var x end metavar + metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar cut lemma distributionOut conclude metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar * metavar var x end metavar + metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar modus ponens metavar var x end metavar * metavar var y end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar conclude metavar var y end metavar * metavar var x end metavar + metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma x+x=2*x as system Q infer all metavar var x end metavar indeed metavar var x end metavar + metavar var x end metavar = 1 + 1 * metavar var x end metavar end define end math ] "


" [ math define proof of lemma x+x=2*x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed axiom times1 conclude metavar var x end metavar * 1 = metavar var x end metavar cut lemma eqSymmetry conclude metavar var x end metavar = metavar var x end metavar * 1 cut lemma eqAdditionLeft modus ponens metavar var x end metavar = metavar var x end metavar * 1 conclude metavar var x end metavar + metavar var x end metavar = metavar var x end metavar + metavar var x end metavar * 1 cut lemma eqAddition modus ponens metavar var x end metavar = metavar var x end metavar * 1 conclude metavar var x end metavar + metavar var x end metavar * 1 = metavar var x end metavar * 1 + metavar var x end metavar * 1 cut lemma eqTransitivity modus ponens metavar var x end metavar + metavar var x end metavar = metavar var x end metavar + metavar var x end metavar * 1 modus ponens metavar var x end metavar + metavar var x end metavar * 1 = metavar var x end metavar * 1 + metavar var x end metavar * 1 conclude metavar var x end metavar + metavar var x end metavar = metavar var x end metavar * 1 + metavar var x end metavar * 1 cut lemma distributionOut conclude metavar var x end metavar * 1 + metavar var x end metavar * 1 = metavar var x end metavar * 1 + 1 cut 1rule repetition modus ponens metavar var x end metavar * 1 + metavar var x end metavar * 1 = metavar var x end metavar * 1 + 1 conclude metavar var x end metavar * 1 + metavar var x end metavar * 1 = metavar var x end metavar * 1 + 1 cut axiom timesCommutativity conclude metavar var x end metavar * 1 + 1 = 1 + 1 * metavar var x end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar + metavar var x end metavar = metavar var x end metavar * 1 + metavar var x end metavar * 1 modus ponens metavar var x end metavar * 1 + metavar var x end metavar * 1 = metavar var x end metavar * 1 + 1 modus ponens metavar var x end metavar * 1 + 1 = 1 + 1 * metavar var x end metavar conclude metavar var x end metavar + metavar var x end metavar = 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma (1/2)x+(1/2)x=x as system Q infer all metavar var x end metavar indeed 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma (1/2)x+(1/2)x=x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma 0<2 conclude not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 cut lemma lessNeq modus ponens not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 conclude not0 0 = 1 + 1 cut lemma neqSymmetry modus ponens not0 0 = 1 + 1 conclude not0 1 + 1 = 0 cut lemma x+x=2*x conclude 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar cut axiom timesAssociativity conclude 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar conclude 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar cut lemma reciprocal modus ponens not0 1 + 1 = 0 conclude 1 + 1 * 1/ 1 + 1 = 1 cut lemma eqMultiplication modus ponens 1 + 1 * 1/ 1 + 1 = 1 conclude 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 * metavar var x end metavar cut lemma times1Left conclude 1 * metavar var x end metavar = metavar var x end metavar cut lemma eqTransitivity5 modus ponens 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar modus ponens 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 * metavar var x end metavar modus ponens 1 + 1 * 1/ 1 + 1 * metavar var x end metavar = 1 * metavar var x end metavar modus ponens 1 * metavar var x end metavar = metavar var x end metavar conclude 1/ 1 + 1 * metavar var x end metavar + 1/ 1 + 1 * metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x+x+x=3*x as system Q infer all metavar var x end metavar indeed metavar var x end metavar + metavar var x end metavar + metavar var x end metavar = 1 + 1 + 1 * metavar var x end metavar end define end math ] "

" [ math define proof of lemma x+x+x=3*x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma x+x=2*x conclude metavar var x end metavar + metavar var x end metavar = 1 + 1 * metavar var x end metavar cut lemma times1Left conclude 1 * metavar var x end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens 1 * metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar = 1 * metavar var x end metavar cut lemma addEquations modus ponens metavar var x end metavar + metavar var x end metavar = 1 + 1 * metavar var x end metavar modus ponens metavar var x end metavar = 1 * metavar var x end metavar conclude metavar var x end metavar + metavar var x end metavar + metavar var x end metavar = 1 + 1 * metavar var x end metavar + 1 * metavar var x end metavar cut lemma distributionOutLeft conclude 1 + 1 * metavar var x end metavar + 1 * metavar var x end metavar = metavar var x end metavar * 1 + 1 + 1 cut axiom timesCommutativity conclude metavar var x end metavar * 1 + 1 + 1 = 1 + 1 + 1 * metavar var x end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar + metavar var x end metavar + metavar var x end metavar = 1 + 1 * metavar var x end metavar + 1 * metavar var x end metavar modus ponens 1 + 1 * metavar var x end metavar + 1 * metavar var x end metavar = metavar var x end metavar * 1 + 1 + 1 modus ponens metavar var x end metavar * 1 + 1 + 1 = 1 + 1 + 1 * metavar var x end metavar conclude metavar var x end metavar + metavar var x end metavar + metavar var x end metavar = 1 + 1 + 1 * metavar var x end metavar cut 1rule repetition modus ponens metavar var x end metavar + metavar var x end metavar + metavar var x end metavar = 1 + 1 + 1 * metavar var x end metavar conclude metavar var x end metavar + metavar var x end metavar + metavar var x end metavar = 1 + 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma 0<3 as system Q infer not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 end define end math ] "

" [ math define proof of lemma 0<3 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma 0<2 conclude not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 cut lemma lessLeq modus ponens not0 0 <= 1 + 1 imply not0 not0 0 = 1 + 1 conclude 0 <= 1 + 1 cut lemma leqPlus1 modus ponens 0 <= 1 + 1 conclude not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 cut 1rule repetition modus ponens not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 conclude not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma (1/3)x+(1/3)x+(1/3)x=x as system Q infer all metavar var x end metavar indeed 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma (1/3)x+(1/3)x+(1/3)x=x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma 0<3 conclude not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 cut lemma positiveNonzero modus ponens not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 conclude not0 1 + 1 + 1 = 0 cut lemma x+x+x=3*x conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut axiom timesAssociativity conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma reciprocal modus ponens not0 1 + 1 + 1 = 0 conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 = 1 cut lemma eqMultiplication modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 = 1 conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 * metavar var x end metavar cut lemma times1Left conclude 1 * metavar var x end metavar = metavar var x end metavar cut lemma eqTransitivity5 modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 * metavar var x end metavar modus ponens 1 * metavar var x end metavar = metavar var x end metavar conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

XX 0<1/2 er et specialtilfaelde
" [ math define statement of lemma positiveInverted as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= 1/ metavar var x end metavar imply not0 not0 0 = 1/ metavar var x end metavar end define end math ] "

" [ math define proof of lemma positiveInverted as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer prop lemma first conjunct modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut prop lemma second conjunct modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 = metavar var x end metavar cut lemma neqSymmetry modus ponens not0 0 = metavar var x end metavar conclude not0 metavar var x end metavar = 0 cut lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma x*0=0 conclude metavar var x end metavar * 0 = 0 cut lemma x*y=zBackwards modus ponens metavar var x end metavar * 0 = 0 conclude 0 = 0 * metavar var x end metavar cut lemma subLessLeft modus ponens 0 = 0 * metavar var x end metavar modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 0 * metavar var x end metavar <= 1 imply not0 not0 0 * metavar var x end metavar = 1 cut lemma reciprocal modus ponens not0 metavar var x end metavar = 0 conclude metavar var x end metavar * 1/ metavar var x end metavar = 1 cut lemma x*y=zBackwards modus ponens metavar var x end metavar * 1/ metavar var x end metavar = 1 conclude 1 = 1/ metavar var x end metavar * metavar var x end metavar cut lemma subLessRight modus ponens 1 = 1/ metavar var x end metavar * metavar var x end metavar modus ponens not0 0 * metavar var x end metavar <= 1 imply not0 not0 0 * metavar var x end metavar = 1 conclude not0 0 * metavar var x end metavar <= 1/ metavar var x end metavar * metavar var x end metavar imply not0 not0 0 * metavar var x end metavar = 1/ metavar var x end metavar * metavar var x end metavar cut lemma lessDivision modus ponens 0 <= metavar var x end metavar modus ponens not0 0 * metavar var x end metavar <= 1/ metavar var x end metavar * metavar var x end metavar imply not0 not0 0 * metavar var x end metavar = 1/ metavar var x end metavar * metavar var x end metavar conclude not0 0 <= 1/ metavar var x end metavar imply not0 not0 0 = 1/ metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma 0<1/3 as system Q infer not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 end define end math ] "

" [ math define proof of lemma 0<1/3 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma 0<3 conclude not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 cut lemma positiveInverted modus ponens not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 conclude not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 end quote state proof state cache var c end expand end define end math ] "









(*** KVANTI ***)

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

\item " [ math define macro of Nat( var x ) as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "

\item " [ math define macro of meta-sub var a is var b where var x is var t end sub as lambda var t dot lambda var s dot lambda var c dot macro define four var t state var s cache var c definition quote 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 quote end define end define end math ] "

\item " [ math define value of 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 tagged if tagged if var b term root equal quote for all objects var u indeed var v end quote then var b first term equal var x else false end if then var a term equal var b else newline tagged if var b term equal var x then var a term equal var t else tagged if newline var a term root equal var b then meta-sub* var a tail is var b tail where var x is var t end sub else false end if end if end if end define end math ] "

\item " [ math define value of 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 tagged if meta-sub1 var a head is var b head where var x is var t end sub then meta-sub* var a tail is var b tail where var x is var t end sub else false end if end if end define end math ] "

\end{list}








" [ math define statement of lemma fromNotSameF(Weak)(Helper) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar end define end math ] "

" [ math define proof of lemma fromNotSameF(Weak)(Helper) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | infer 0 <= metavar var x end metavar + - metavar var y end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar conclude | metavar var x end metavar + - metavar var y end metavar | = metavar var x end metavar + - metavar var y end metavar cut lemma subLeqRight modus ponens | metavar var x end metavar + - metavar var y end metavar | = metavar var x end metavar + - metavar var y end metavar modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | conclude metavar var z end metavar <= metavar var x end metavar + - metavar var y end metavar cut lemma negativeToLeft(Leq) modus ponens metavar var z end metavar <= metavar var x end metavar + - metavar var y end metavar conclude metavar var z end metavar + metavar var y end metavar <= metavar var x end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var z end metavar + metavar var y end metavar <= metavar var x end metavar conclude metavar var y end metavar + metavar var z end metavar <= metavar var x end metavar cut lemma positiveToRight(Leq) modus ponens metavar var y end metavar + metavar var z end metavar <= metavar var x end metavar conclude metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut prop lemma weaken or first modus ponens metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | infer not0 0 <= metavar var x end metavar + - metavar var y end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar + - metavar var y end metavar conclude not0 metavar var x end metavar + - metavar var y end metavar <= 0 imply not0 not0 metavar var x end metavar + - metavar var y end metavar = 0 cut lemma negativeNumerical conclude | metavar var x end metavar + - metavar var y end metavar | = - metavar var x end metavar + - metavar var y end metavar cut lemma minusNegated conclude - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar cut lemma eqTransitivity modus ponens | metavar var x end metavar + - metavar var y end metavar | = - metavar var x end metavar + - metavar var y end metavar modus ponens - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar conclude | metavar var x end metavar + - metavar var y end metavar | = metavar var y end metavar + - metavar var x end metavar cut lemma subLeqRight modus ponens | metavar var x end metavar + - metavar var y end metavar | = metavar var y end metavar + - metavar var x end metavar modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | conclude metavar var z end metavar <= metavar var y end metavar + - metavar var x end metavar cut lemma negativeToLeft(Leq) modus ponens metavar var z end metavar <= metavar var y end metavar + - metavar var x end metavar conclude metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar cut lemma positiveToRight(Leq) modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar cut prop lemma weaken or second modus ponens metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | infer 0 <= metavar var x end metavar + - metavar var y end metavar infer not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | imply 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | infer not0 0 <= metavar var x end metavar + - metavar var y end metavar infer not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | imply not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut not0 not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer lemma fromNotLess modus ponens not0 not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar conclude metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | cut 1rule mp modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | imply 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | conclude 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut 1rule mp modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | imply not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | conclude not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut prop lemma from negations modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar modus ponens not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromNotSameF(Weak) as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar end define end math ] "

" [ math define proof of lemma fromNotSameF(Weak) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule deduction modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut pred lemma AEAnegated modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar infer prop lemma from negated double imply modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut prop lemma first conjunct modus ponens not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar cut prop lemma second conjunct modus ponens not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut lemma fromNotSameF(Weak)(Helper) modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar cut prop lemma join conjuncts modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar conclude not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar infer not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar imply not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar cut pred lemma addEAE modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar imply not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar imply not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar cut 1rule mp modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar imply not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma fromMax(1) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar end define end math ] "

" [ math define proof of lemma fromMax(1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer axiom max conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar imply not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar cut prop lemma add double neg modus ponens metavar var y end metavar <= metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar cut prop lemma to negated and(1) modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar conclude not0 not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar cut prop lemma negate second disjunct modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar imply not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar modus ponens not0 not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar cut prop lemma second conjunct modus ponens not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar conclude if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromMax(2) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar <= metavar var x end metavar infer if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar end define end math ] "

" [ math define proof of lemma fromMax(2) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar <= metavar var x end metavar infer axiom max conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar imply not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar cut prop lemma to negated and(1) modus ponens not0 metavar var y end metavar <= metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar cut prop lemma negate first disjunct modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar imply not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar conclude not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar cut prop lemma second conjunct modus ponens not0 not0 metavar var y end metavar <= metavar var x end metavar imply not0 if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar conclude if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma leqMax1 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) end define end math ] "

" [ math define proof of lemma leqMax1 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer lemma fromMax(1) modus ponens metavar var y end metavar <= metavar var x end metavar conclude if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar cut lemma eqSymmetry modus ponens if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar conclude metavar var x end metavar = if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut lemma eqLeq modus ponens metavar var x end metavar = if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) conclude metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar <= metavar var x end metavar infer lemma fromMax(2) modus ponens not0 metavar var y end metavar <= metavar var x end metavar conclude if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar cut lemma eqSymmetry modus ponens if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar conclude metavar var y end metavar = if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut lemma toLess modus ponens not0 metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut lemma lessLeq modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar cut lemma subLeqRight modus ponens metavar var y end metavar = if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) conclude metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar <= metavar var x end metavar infer metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) conclude not0 metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut prop lemma from negations modus ponens metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) modus ponens not0 metavar var y end metavar <= metavar var x end metavar imply metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) conclude metavar var x end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma leqMax2 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) end define end math ] "

" [ math define proof of lemma leqMax2 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer lemma fromMax(1) modus ponens metavar var y end metavar <= metavar var x end metavar conclude if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar cut lemma eqSymmetry modus ponens if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var x end metavar conclude metavar var x end metavar = if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut lemma subLeqRight modus ponens metavar var x end metavar = if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) modus ponens metavar var y end metavar <= metavar var x end metavar conclude metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar <= metavar var x end metavar infer lemma fromMax(2) modus ponens not0 metavar var y end metavar <= metavar var x end metavar conclude if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar cut lemma eqSymmetry modus ponens if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) = metavar var y end metavar conclude metavar var y end metavar = if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut lemma eqLeq modus ponens metavar var y end metavar = if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) conclude metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar <= metavar var x end metavar infer metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) conclude metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var y end metavar <= metavar var x end metavar infer metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) conclude not0 metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) cut prop lemma from negations modus ponens metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) modus ponens not0 metavar var y end metavar <= metavar var x end metavar imply metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) conclude metavar var y end metavar <= if( metavar var y end metavar <= metavar var x end metavar , metavar var x end metavar , metavar var y end metavar ) end quote state proof state cache var c end expand end define end math ] "






" [ math define statement of lemma negativeToRight(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar infer metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma negativeToRight(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar infer lemma leqAddition modus ponens metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar conclude metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar <= metavar var z end metavar + metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma three2threeTerms conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar conclude metavar var x end metavar = metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar conclude metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar modus ponens metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar <= metavar var z end metavar + metavar var y end metavar conclude metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma lessThanMax as system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var a end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var a end metavar imply not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar infer not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) end define end math ] "

" [ math define proof of lemma lessThanMax as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var a end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var a end metavar infer 1rule mp modus ponens metavar var a end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var a end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut lemma leqMax1 conclude metavar var y end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) cut lemma lessLeqTransitivity modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) conclude not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) cut all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var a end metavar imply not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar infer not0 metavar var a end metavar infer 1rule mp modus ponens not0 metavar var a end metavar imply not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar modus ponens not0 metavar var a end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar cut lemma leqMax2 conclude metavar var z end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) cut lemma lessLeqTransitivity modus ponens not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar modus ponens metavar var z end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) conclude not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) cut all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var a end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer metavar var a end metavar infer not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) conclude metavar var a end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var a end metavar imply not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) cut 1rule deduction modus ponens all metavar var a end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var a end metavar imply not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar infer not0 metavar var a end metavar infer not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) conclude not0 metavar var a end metavar imply not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar imply not0 metavar var a end metavar imply not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) cut metavar var a end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 metavar var a end metavar imply not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar infer 1rule mp modus ponens metavar var a end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply metavar var a end metavar imply not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) modus ponens metavar var a end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude metavar var a end metavar imply not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) cut 1rule mp modus ponens not0 metavar var a end metavar imply not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar imply not0 metavar var a end metavar imply not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) modus ponens not0 metavar var a end metavar imply not0 metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar conclude not0 metavar var a end metavar imply not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) cut prop lemma from negations modus ponens metavar var a end metavar imply not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) modus ponens not0 metavar var a end metavar imply not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) conclude not0 metavar var x end metavar <= if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) imply not0 not0 metavar var x end metavar = if( metavar var z end metavar <= metavar var y end metavar , metavar var y end metavar , metavar var z end metavar ) end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma nonnegativeFactors as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer 0 <= metavar var x end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma nonnegativeFactors as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma leqMultiplication modus ponens 0 <= metavar var y end metavar modus ponens 0 <= metavar var x end metavar conclude 0 * metavar var y end metavar <= metavar var x end metavar * metavar var y end metavar cut axiom timesCommutativity conclude 0 * metavar var y end metavar = metavar var y end metavar * 0 cut lemma x*0=0 conclude metavar var y end metavar * 0 = 0 cut lemma eqTransitivity modus ponens 0 * metavar var y end metavar = metavar var y end metavar * 0 modus ponens metavar var y end metavar * 0 = 0 conclude 0 * metavar var y end metavar = 0 cut lemma subLeqLeft modus ponens 0 * metavar var y end metavar = 0 modus ponens 0 * metavar var y end metavar <= metavar var x end metavar * metavar var y end metavar conclude 0 <= metavar var x end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma multiplyEquations as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar = metavar var u end metavar infer metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var u end metavar end define end math ] "

" [ math define proof of lemma multiplyEquations as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var z end metavar = metavar var u end metavar infer lemma eqMultiplication modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar cut lemma eqMultiplicationLeft modus ponens metavar var z end metavar = metavar var u end metavar conclude metavar var y end metavar * metavar var z end metavar = metavar var y end metavar * metavar var u end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar = metavar var y end metavar * metavar var u end metavar conclude metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma minusTimesMinus as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed - metavar var x end metavar * - metavar var y end metavar = metavar var x end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma minusTimesMinus as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma times(-1)Left conclude - 1 * - metavar var y end metavar = - - metavar var y end metavar cut lemma eqTransitivity modus ponens - 1 * - metavar var y end metavar = - - metavar var y end metavar modus ponens - - metavar var y end metavar = metavar var y end metavar conclude - 1 * - metavar var y end metavar = metavar var y end metavar cut lemma eqMultiplicationLeft modus ponens - 1 * - metavar var y end metavar = metavar var y end metavar conclude metavar var x end metavar * - 1 * - metavar var y end metavar = metavar var x end metavar * metavar var y end metavar cut lemma times(-1) conclude metavar var x end metavar * - 1 = - metavar var x end metavar cut lemma eqMultiplication modus ponens metavar var x end metavar * - 1 = - metavar var x end metavar conclude metavar var x end metavar * - 1 * - metavar var y end metavar = - metavar var x end metavar * - metavar var y end metavar cut axiom timesAssociativity conclude metavar var x end metavar * - 1 * - metavar var y end metavar = metavar var x end metavar * - 1 * - metavar var y end metavar cut lemma equality modus ponens metavar var x end metavar * - 1 * - metavar var y end metavar = - metavar var x end metavar * - metavar var y end metavar modus ponens metavar var x end metavar * - 1 * - metavar var y end metavar = metavar var x end metavar * - 1 * - metavar var y end metavar conclude - metavar var x end metavar * - metavar var y end metavar = metavar var x end metavar * - 1 * - metavar var y end metavar cut lemma eqTransitivity modus ponens - metavar var x end metavar * - metavar var y end metavar = metavar var x end metavar * - 1 * - metavar var y end metavar modus ponens metavar var x end metavar * - 1 * - metavar var y end metavar = metavar var x end metavar * metavar var y end metavar conclude - metavar var x end metavar * - metavar var y end metavar = metavar var x end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma plusTimesMinus as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar * - metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma plusTimesMinus as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma times(-1)Left conclude - 1 * metavar var y end metavar = - metavar var y end metavar cut lemma eqMultiplicationLeft modus ponens - 1 * metavar var y end metavar = - metavar var y end metavar conclude metavar var x end metavar * - 1 * metavar var y end metavar = metavar var x end metavar * - metavar var y end metavar cut axiom timesAssociativity conclude metavar var x end metavar * - 1 * metavar var y end metavar = metavar var x end metavar * - 1 * metavar var y end metavar cut axiom timesCommutativity conclude metavar var x end metavar * - 1 = - 1 * metavar var x end metavar cut lemma eqMultiplication modus ponens metavar var x end metavar * - 1 = - 1 * metavar var x end metavar conclude metavar var x end metavar * - 1 * metavar var y end metavar = - 1 * metavar var x end metavar * metavar var y end metavar cut axiom timesAssociativity conclude - 1 * metavar var x end metavar * metavar var y end metavar = - 1 * metavar var x end metavar * metavar var y end metavar cut lemma times(-1)Left conclude - 1 * metavar var x end metavar * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar * - 1 * metavar var y end metavar = - 1 * metavar var x end metavar * metavar var y end metavar modus ponens - 1 * metavar var x end metavar * metavar var y end metavar = - 1 * metavar var x end metavar * metavar var y end metavar modus ponens - 1 * metavar var x end metavar * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar conclude metavar var x end metavar * - 1 * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar cut lemma equality modus ponens metavar var x end metavar * - 1 * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar modus ponens metavar var x end metavar * - 1 * metavar var y end metavar = metavar var x end metavar * - 1 * metavar var y end metavar conclude - metavar var x end metavar * metavar var y end metavar = metavar var x end metavar * - 1 * metavar var y end metavar cut lemma eqTransitivity modus ponens - metavar var x end metavar * metavar var y end metavar = metavar var x end metavar * - 1 * metavar var y end metavar modus ponens metavar var x end metavar * - 1 * metavar var y end metavar = metavar var x end metavar * - metavar var y end metavar conclude - metavar var x end metavar * metavar var y end metavar = metavar var x end metavar * - metavar var y end metavar cut lemma eqSymmetry modus ponens - metavar var x end metavar * metavar var y end metavar = metavar var x end metavar * - metavar var y end metavar conclude metavar var x end metavar * - metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma splitNumericalProduct(++) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalProduct(++) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma nonnegativeFactors modus ponens 0 <= metavar var x end metavar modus ponens 0 <= metavar var y end metavar conclude 0 <= metavar var x end metavar * metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar * metavar var y end metavar conclude | metavar var x end metavar * metavar var y end metavar | = metavar var x end metavar * metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var y end metavar conclude | metavar var y end metavar | = metavar var y end metavar cut lemma multiplyEquations modus ponens | metavar var x end metavar | = metavar var x end metavar modus ponens | metavar var y end metavar | = metavar var y end metavar conclude | metavar var x end metavar | * | metavar var y end metavar | = metavar var x end metavar * metavar var y end metavar cut lemma eqSymmetry modus ponens | metavar var x end metavar | * | metavar var y end metavar | = metavar var x end metavar * metavar var y end metavar conclude metavar var x end metavar * metavar var y end metavar = | metavar var x end metavar | * | metavar var y end metavar | cut lemma eqTransitivity modus ponens | metavar var x end metavar * metavar var y end metavar | = metavar var x end metavar * metavar var y end metavar modus ponens metavar var x end metavar * metavar var y end metavar = | metavar var x end metavar | * | metavar var y end metavar | conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma splitNumericalProduct(+-) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | end define end math ] "

" [ math define proof of lemma splitNumericalProduct(+-) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma signNumerical conclude | metavar var x end metavar * metavar var y end metavar | = | - metavar var x end metavar * metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar * metavar var y end metavar | = | - metavar var x end metavar * metavar var y end metavar | conclude | - metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar * metavar var y end metavar | cut lemma plusTimesMinus conclude metavar var x end metavar * - metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar cut lemma sameNumerical modus ponens metavar var x end metavar * - metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar conclude | metavar var x end metavar * - metavar var y end metavar | = | - metavar var x end metavar * metavar var y end metavar | cut lemma eqTransitivity modus ponens | metavar var x end metavar * - metavar var y end metavar | = | - metavar var x end metavar * metavar var y end metavar | modus ponens | - metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar * metavar var y end metavar | conclude | metavar var x end metavar * - metavar var y end metavar | = | metavar var x end metavar * metavar var y end metavar | cut lemma signNumerical conclude | metavar var y end metavar | = | - metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var y end metavar | = | - metavar var y end metavar | conclude | - metavar var y end metavar | = | metavar var y end metavar | cut lemma eqMultiplicationLeft modus ponens | - metavar var y end metavar | = | metavar var y end metavar | conclude | metavar var x end metavar | * | - metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma splitNumericalProduct(++) modus ponens 0 <= metavar var x end metavar modus ponens 0 <= - metavar var y end metavar conclude | metavar var x end metavar * - metavar var y end metavar | = | metavar var x end metavar | * | - metavar var y end metavar | cut lemma eqTransitivity modus ponens | metavar var x end metavar * - metavar var y end metavar | = | metavar var x end metavar | * | - metavar var y end metavar | modus ponens | metavar var x end metavar | * | - metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude | metavar var x end metavar * - metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma equality modus ponens | metavar var x end metavar * - metavar var y end metavar | = | metavar var x end metavar * metavar var y end metavar | modus ponens | metavar var x end metavar * - metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma splitNumericalProduct as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | end define end math ] "


" [ math define proof of lemma splitNumericalProduct as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma splitNumericalProduct(++) modus ponens 0 <= metavar var x end metavar modus ponens 0 <= metavar var y end metavar conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalProduct(+-) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer lemma splitNumericalProduct(+-) modus ponens 0 <= metavar var y end metavar modus ponens metavar var x end metavar <= 0 conclude | metavar var y end metavar * metavar var x end metavar | = | metavar var y end metavar | * | metavar var x end metavar | cut axiom timesCommutativity conclude metavar var x end metavar * metavar var y end metavar = metavar var y end metavar * metavar var x end metavar cut lemma sameNumerical modus ponens metavar var x end metavar * metavar var y end metavar = metavar var y end metavar * metavar var x end metavar conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var y end metavar * metavar var x end metavar | cut axiom timesCommutativity conclude | metavar var y end metavar | * | metavar var x end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma eqTransitivity4 modus ponens | metavar var x end metavar * metavar var y end metavar | = | metavar var y end metavar * metavar var x end metavar | modus ponens | metavar var y end metavar * metavar var x end metavar | = | metavar var y end metavar | * | metavar var x end metavar | modus ponens | metavar var y end metavar | * | metavar var x end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer lemma nonpositiveNegated modus ponens metavar var x end metavar <= 0 conclude 0 <= - metavar var x end metavar cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma splitNumericalProduct(++) modus ponens 0 <= - metavar var x end metavar modus ponens 0 <= - metavar var y end metavar conclude | - metavar var x end metavar * - metavar var y end metavar | = | - metavar var x end metavar | * | - metavar var y end metavar | cut lemma minusTimesMinus conclude - metavar var x end metavar * - metavar var y end metavar = metavar var x end metavar * metavar var y end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar * - metavar var y end metavar = metavar var x end metavar * metavar var y end metavar conclude | - metavar var x end metavar * - metavar var y end metavar | = | metavar var x end metavar * metavar var y end metavar | cut lemma eqSymmetry modus ponens | - metavar var x end metavar * - metavar var y end metavar | = | metavar var x end metavar * metavar var y end metavar | conclude | metavar var x end metavar * metavar var y end metavar | = | - metavar var x end metavar * - metavar var y end metavar | cut lemma signNumerical conclude | metavar var x end metavar | = | - metavar var x end metavar | cut lemma signNumerical conclude | metavar var y end metavar | = | - metavar var y end metavar | cut lemma multiplyEquations modus ponens | metavar var x end metavar | = | - metavar var x end metavar | modus ponens | metavar var y end metavar | = | - metavar var y end metavar | conclude | metavar var x end metavar | * | metavar var y end metavar | = | - metavar var x end metavar | * | - metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar | * | metavar var y end metavar | = | - metavar var x end metavar | * | - metavar var y end metavar | conclude | - metavar var x end metavar | * | - metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma eqTransitivity4 modus ponens | metavar var x end metavar * metavar var y end metavar | = | - metavar var x end metavar * - metavar var y end metavar | modus ponens | - metavar var x end metavar * - metavar var y end metavar | = | - metavar var x end metavar | * | - metavar var y end metavar | modus ponens | - metavar var x end metavar | * | - metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude metavar var x end metavar <= 0 imply metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | modus ponens metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | modus ponens metavar var x end metavar <= 0 imply metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | modus ponens metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma three2threeFactors as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma three2threeFactors as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed axiom timesCommutativity conclude metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar cut lemma three2twoFactors modus ponens metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar conclude metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar * metavar var y end metavar cut axiom timesAssociativity conclude metavar var x end metavar * metavar var z end metavar * metavar var y end metavar = metavar var x end metavar * metavar var z end metavar * metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar * metavar var z end metavar * metavar var y end metavar = metavar var x end metavar * metavar var z end metavar * metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar * metavar var y end metavar = metavar var x end metavar * metavar var z end metavar * metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar * metavar var y end metavar modus ponens metavar var x end metavar * metavar var z end metavar * metavar var y end metavar = metavar var x end metavar * metavar var z end metavar * metavar var y end metavar conclude metavar var x end metavar * metavar var y end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma nonzeroFactors as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = 0 infer not0 metavar var y end metavar = 0 infer not0 metavar var x end metavar * metavar var y end metavar = 0 end define end math ] "

" [ math define proof of lemma nonzeroFactors as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = 0 infer not0 metavar var y end metavar = 0 infer lemma neqMultiplication modus ponens not0 metavar var y end metavar = 0 modus ponens not0 metavar var x end metavar = 0 conclude not0 metavar var x end metavar * metavar var y end metavar = 0 * metavar var y end metavar cut axiom timesCommutativity conclude 0 * metavar var y end metavar = metavar var y end metavar * 0 cut lemma x*0=0 conclude metavar var y end metavar * 0 = 0 cut lemma eqTransitivity modus ponens 0 * metavar var y end metavar = metavar var y end metavar * 0 modus ponens metavar var y end metavar * 0 = 0 conclude 0 * metavar var y end metavar = 0 cut lemma subNeqRight modus ponens 0 * metavar var y end metavar = 0 modus ponens not0 metavar var x end metavar * metavar var y end metavar = 0 * metavar var y end metavar conclude not0 metavar var x end metavar * metavar var y end metavar = 0 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveFactors as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= metavar var y end metavar imply not0 not0 0 = metavar var y end metavar infer not0 0 <= metavar var x end metavar * metavar var y end metavar imply not0 not0 0 = metavar var x end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma positiveFactors as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= metavar var y end metavar imply not0 not0 0 = metavar var y end metavar infer 1rule repetition modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar cut prop lemma first conjunct modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut prop lemma second conjunct modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 = metavar var x end metavar cut lemma neqSymmetry modus ponens not0 0 = metavar var x end metavar conclude not0 metavar var x end metavar = 0 cut 1rule repetition modus ponens not0 0 <= metavar var y end metavar imply not0 not0 0 = metavar var y end metavar conclude not0 0 <= metavar var y end metavar imply not0 not0 0 = metavar var y end metavar cut prop lemma first conjunct modus ponens not0 0 <= metavar var y end metavar imply not0 not0 0 = metavar var y end metavar conclude 0 <= metavar var y end metavar cut prop lemma second conjunct modus ponens not0 0 <= metavar var y end metavar imply not0 not0 0 = metavar var y end metavar conclude not0 0 = metavar var y end metavar cut lemma neqSymmetry modus ponens not0 0 = metavar var y end metavar conclude not0 metavar var y end metavar = 0 cut lemma nonnegativeFactors modus ponens 0 <= metavar var x end metavar modus ponens 0 <= metavar var y end metavar conclude 0 <= metavar var x end metavar * metavar var y end metavar cut lemma nonzeroFactors modus ponens not0 metavar var x end metavar = 0 modus ponens not0 metavar var y end metavar = 0 conclude not0 metavar var x end metavar * metavar var y end metavar = 0 cut lemma neqSymmetry modus ponens not0 metavar var x end metavar * metavar var y end metavar = 0 conclude not0 0 = metavar var x end metavar * metavar var y end metavar cut prop lemma join conjuncts modus ponens 0 <= metavar var x end metavar * metavar var y end metavar modus ponens not0 0 = metavar var x end metavar * metavar var y end metavar conclude not0 0 <= metavar var x end metavar * metavar var y end metavar imply not0 not0 0 = metavar var x end metavar * metavar var y end metavar cut 1rule repetition modus ponens not0 0 <= metavar var x end metavar * metavar var y end metavar imply not0 not0 0 = metavar var x end metavar * metavar var y end metavar conclude not0 0 <= metavar var x end metavar * metavar var y end metavar imply not0 not0 0 = metavar var x end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma seriesSubsetCP as system Q infer all metavar var fx end metavar indeed all metavar var sy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end define end math ] "

" [ math define proof of lemma seriesSubsetCP as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var sy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer object var var s1 end var in0 metavar var fx end metavar infer lemma fromSeries modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut prop lemma first conjunct modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut prop lemma first conjunct modus ponens not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var conclude for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma a4 at object var var s1 end var modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude object var var s1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule mp modus ponens object var var s1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens object var var s1 end var in0 metavar var fx end metavar conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut all metavar var sy end metavar indeed not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer prop lemma first conjunct modus ponens not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar cut prop lemma first conjunct modus ponens not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar conclude object var var op1 end var in0 N cut prop lemma second conjunct modus ponens not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar conclude object var var op2 end var in0 metavar var sy end metavar cut prop lemma second conjunct modus ponens not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma eqSymmetry modus ponens object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair = object var var s1 end var cut lemma toCartProd modus ponens object var var op1 end var in0 N modus ponens object var var op2 end var in0 metavar var sy end metavar conclude zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma sameMember modus ponens zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair = object var var s1 end var modus ponens zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut 1rule deduction modus ponens all metavar var sy end metavar indeed not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut pred lemma 2exist mp modus ponens not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var s1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut all metavar var fx end metavar indeed all metavar var sy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var sy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer object var var s1 end var in0 metavar var fx end metavar infer object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar imply for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer 1rule mp modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar imply for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma fromCartProd as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar end define end math ] "

" [ math define proof of lemma fromCartProd as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer 1rule repetition modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma separation2formula modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power imply not0 not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut prop lemma second conjunct modus ponens not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power imply not0 not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule repetition modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer prop lemma first conjunct modus ponens not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar cut prop lemma first conjunct modus ponens not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar conclude object var var op1 end var in0 metavar var sx1 end metavar cut prop lemma second conjunct modus ponens not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar conclude object var var op2 end var in0 metavar var sy1 end metavar cut prop lemma second conjunct modus ponens not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude metavar var sx end metavar = object var var op1 end var cut lemma sameMember(2) modus ponens metavar var sx end metavar = object var var op1 end var modus ponens object var var op1 end var in0 metavar var sx1 end metavar conclude metavar var sx end metavar in0 metavar var sx1 end metavar cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude metavar var sy end metavar = object var var op2 end var cut lemma sameMember(2) modus ponens metavar var sy end metavar = object var var op2 end var modus ponens object var var op2 end var in0 metavar var sy1 end metavar conclude metavar var sy end metavar in0 metavar var sy1 end metavar cut prop lemma join conjuncts modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar conclude not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar cut 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar conclude not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar cut pred lemma 2exist mp modus ponens not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma fromCartProd(1) as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer metavar var sx end metavar in0 metavar var sx1 end metavar end define end math ] "

" [ math define proof of lemma fromCartProd(1) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer lemma fromCartProd modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar cut prop lemma first conjunct modus ponens not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar conclude metavar var sx end metavar in0 metavar var sx1 end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromCartProd(2) as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer metavar var sy end metavar in0 metavar var sy1 end metavar end define end math ] "

" [ math define proof of lemma fromCartProd(2) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set infer lemma fromCartProd modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair in0 the set of ph in power power U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 metavar var sx1 end metavar imply not0 object var var op2 end var in0 metavar var sy1 end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar cut prop lemma second conjunct modus ponens not0 metavar var sx end metavar in0 metavar var sx1 end metavar imply not0 metavar var sy end metavar in0 metavar var sy1 end metavar conclude metavar var sy end metavar in0 metavar var sy1 end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma valueType as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var sy end metavar indeed metavar var m end metavar in0 N infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer [ metavar var fx end metavar ; metavar var m end metavar ] in0 metavar var sy end metavar end define end math ] "

" [ math define proof of lemma valueType as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var sy end metavar indeed metavar var m end metavar in0 N infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer lemma memberOfSeries modus ponens metavar var m end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair in0 metavar var fx end metavar cut lemma seriesSubsetCP modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma fromSubset modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair in0 metavar var fx end metavar conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma fromCartProd(2) modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma metavar var sy end metavar end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sy end metavar imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set conclude [ metavar var fx end metavar ; metavar var m end metavar ] in0 metavar var sy end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma productIsFunction as system Q infer all metavar var m1 end metavar indeed all metavar var m2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var end define end math ] "

" [ math define proof of lemma productIsFunction as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m1 end metavar indeed all metavar var m2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed object var var f1 end var = object var var f3 end var infer zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma [ metavar var fx end metavar ; metavar var m1 end metavar ] * [ metavar var fy end metavar ; metavar var m1 end metavar ] end pair end pair infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m2 end metavar comma metavar var m2 end metavar end pair comma zermelo pair metavar var m2 end metavar comma [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] end pair end pair infer lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma [ metavar var fx end metavar ; metavar var m1 end metavar ] * [ metavar var fy end metavar ; metavar var m1 end metavar ] end pair end pair conclude object var var f1 end var = metavar var m1 end metavar cut lemma eqSymmetry modus ponens object var var f1 end var = metavar var m1 end metavar conclude metavar var m1 end metavar = object var var f1 end var cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m2 end metavar comma metavar var m2 end metavar end pair comma zermelo pair metavar var m2 end metavar comma [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] end pair end pair conclude object var var f3 end var = metavar var m2 end metavar cut lemma eqTransitivity4 modus ponens metavar var m1 end metavar = object var var f1 end var modus ponens object var var f1 end var = object var var f3 end var modus ponens object var var f3 end var = metavar var m2 end metavar conclude metavar var m1 end metavar = metavar var m2 end metavar cut lemma sameSeries modus ponens metavar var m1 end metavar = metavar var m2 end metavar conclude [ metavar var fx end metavar ; metavar var m1 end metavar ] = [ metavar var fx end metavar ; metavar var m2 end metavar ] cut lemma sameSeries modus ponens metavar var m1 end metavar = metavar var m2 end metavar conclude [ metavar var fy end metavar ; metavar var m1 end metavar ] = [ metavar var fy end metavar ; metavar var m2 end metavar ] cut lemma multiplyEquations modus ponens [ metavar var fx end metavar ; metavar var m1 end metavar ] = [ metavar var fx end metavar ; metavar var m2 end metavar ] modus ponens [ metavar var fy end metavar ; metavar var m1 end metavar ] = [ metavar var fy end metavar ; metavar var m2 end metavar ] conclude [ metavar var fx end metavar ; metavar var m1 end metavar ] * [ metavar var fy end metavar ; metavar var m1 end metavar ] = [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma [ metavar var fx end metavar ; metavar var m1 end metavar ] * [ metavar var fy end metavar ; metavar var m1 end metavar ] end pair end pair conclude object var var f2 end var = [ metavar var fx end metavar ; metavar var m1 end metavar ] * [ metavar var fy end metavar ; metavar var m1 end metavar ] cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m2 end metavar comma metavar var m2 end metavar end pair comma zermelo pair metavar var m2 end metavar comma [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] end pair end pair conclude object var var f4 end var = [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] cut lemma eqSymmetry modus ponens object var var f4 end var = [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] conclude [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] = object var var f4 end var cut lemma eqTransitivity4 modus ponens object var var f2 end var = [ metavar var fx end metavar ; metavar var m1 end metavar ] * [ metavar var fy end metavar ; metavar var m1 end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m1 end metavar ] * [ metavar var fy end metavar ; metavar var m1 end metavar ] = [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] = object var var f4 end var conclude object var var f2 end var = object var var f4 end var cut all metavar var m1 end metavar indeed all metavar var m2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m1 end metavar indeed all metavar var m2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed object var var f1 end var = object var var f3 end var infer zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma [ metavar var fx end metavar ; metavar var m1 end metavar ] * [ metavar var fy end metavar ; metavar var m1 end metavar ] end pair end pair infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m2 end metavar comma metavar var m2 end metavar end pair comma zermelo pair metavar var m2 end metavar comma [ metavar var fx end metavar ; metavar var m2 end metavar ] * [ metavar var fy end metavar ; metavar var m2 end metavar ] end pair end pair infer object var var f2 end var = object var var f4 end var conclude for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma productIsTotal as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set end define end math ] "

" [ math define proof of lemma productIsTotal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed object var var s1 end var in0 N infer axiom seriesType conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut axiom seriesType conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar cut lemma valueType modus ponens object var var s1 end var in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude [ metavar var fx end metavar ; object var var s1 end var ] in0 Q cut lemma valueType modus ponens object var var s1 end var in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar conclude [ metavar var fy end metavar ; object var var s1 end var ] in0 Q cut 1rule Qclosed(Multiplication) modus ponens [ metavar var fx end metavar ; object var var s1 end var ] in0 Q modus ponens [ metavar var fy end metavar ; object var var s1 end var ] in0 Q conclude [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] in0 Q cut lemma toCartProd modus ponens object var var s1 end var in0 N modus ponens [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] in0 Q conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma eqReflexivity conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair cut pred lemma intro exist at object var var s1 end var modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair conclude not0 for all objects metavar var m end metavar indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair cut lemma formula2separation modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 for all objects metavar var m end metavar indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set cut 1rule repetition modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set cut pred lemma intro exist at [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma [ metavar var fx end metavar ; object var var s1 end var ] * [ metavar var fy end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set conclude not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set cut all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed object var var s1 end var in0 N infer not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma productIsRationalSeries as system Q infer all metavar var m end metavar indeed all metavar var m1 end metavar indeed all metavar var m2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set end define end math ] "

" [ math define proof of lemma productIsRationalSeries as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var m1 end metavar indeed all metavar var m2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma CPseparationIsRelation conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma productIsFunction conclude for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut lemma productIsTotal conclude for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set cut lemma toSeries modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma timesF as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed lambda var c dot typeNat0( quote metavar var m end metavar end quote ) endorse lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) endorse lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) endorse [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end define end math ] "

" [ math define proof of lemma timesF as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed lambda var c dot typeNat0( quote metavar var m end metavar end quote ) endorse lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) endorse lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) endorse axiom natType conclude metavar var m end metavar in0 N cut axiom seriesType conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut axiom seriesType conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar cut lemma productIsRationalSeries conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set cut lemma memberOfSeries modus ponens metavar var m end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set cut lemma valueType modus ponens metavar var m end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] in0 Q cut lemma valueType modus ponens metavar var m end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar conclude [ metavar var fy end metavar ; metavar var m end metavar ] in0 Q cut 1rule Qclosed(Multiplication) modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] in0 Q modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] in0 Q conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] in0 Q cut lemma toCartProd modus ponens metavar var m end metavar in0 N modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] in0 Q conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma eqReflexivity conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair cut pred lemma intro exist at metavar var m end metavar modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair conclude not0 for all objects metavar var m end metavar indeed not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair cut lemma formula2separation modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 for all objects metavar var m end metavar indeed not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set cut lemma eqReflexivity conclude metavar var m end metavar = metavar var m end metavar cut lemma uniqueMember modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set modus ponens metavar var m end metavar = metavar var m end metavar conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma timesF(Sym) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end define end math ] "

" [ math define proof of lemma timesF(Sym) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed lemma timesF conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma sameOrderedPair as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar = metavar var sx1 end metavar infer metavar var sy end metavar = metavar var sy1 end metavar infer zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair end define end math ] "

" [ math define proof of lemma sameOrderedPair as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar = metavar var sx1 end metavar infer metavar var sy end metavar = metavar var sy1 end metavar infer lemma same singleton modus ponens metavar var sx end metavar = metavar var sx1 end metavar conclude zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair cut lemma same pair modus ponens metavar var sx end metavar = metavar var sx1 end metavar modus ponens metavar var sy end metavar = metavar var sy1 end metavar conclude zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair cut lemma same pair modus ponens zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair modus ponens zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair = zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair cut 1rule repetition modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair conclude zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair = zermelo pair zermelo pair metavar var sx1 end metavar comma metavar var sx1 end metavar end pair comma zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair end pair end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma inSeries helper as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sy end metavar in0 metavar var fx end metavar imply not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar imply not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end define end math ] "

" [ math define proof of lemma inSeries helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sy end metavar in0 metavar var fx end metavar infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer prop lemma first conjunct modus ponens not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar cut prop lemma first conjunct modus ponens not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar conclude object var var op1 end var in0 N cut prop lemma second conjunct modus ponens not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma sameMember modus ponens metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens metavar var sy end metavar in0 metavar var fx end metavar conclude zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair in0 metavar var fx end metavar cut lemma memberOfSeries modus ponens object var var op1 end var in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma [ metavar var fx end metavar ; object var var op1 end var ] end pair end pair in0 metavar var fx end metavar cut lemma eqReflexivity conclude object var var op1 end var = object var var op1 end var cut lemma uniqueMember modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar modus ponens zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair in0 metavar var fx end metavar modus ponens zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma [ metavar var fx end metavar ; object var var op1 end var ] end pair end pair in0 metavar var fx end metavar modus ponens object var var op1 end var = object var var op1 end var conclude object var var op2 end var = [ metavar var fx end metavar ; object var var op1 end var ] cut lemma sameOrderedPair modus ponens object var var op1 end var = object var var op1 end var modus ponens object var var op2 end var = [ metavar var fx end metavar ; object var var op1 end var ] conclude zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma [ metavar var fx end metavar ; object var var op1 end var ] end pair end pair cut lemma eqTransitivity modus ponens metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma [ metavar var fx end metavar ; object var var op1 end var ] end pair end pair conclude metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma [ metavar var fx end metavar ; object var var op1 end var ] end pair end pair cut pred lemma intro exist at object var var op1 end var modus ponens metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma [ metavar var fx end metavar ; object var var op1 end var ] end pair end pair conclude not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair cut all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed metavar var sy end metavar in0 metavar var fx end metavar infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair infer not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude metavar var sy end metavar in0 metavar var fx end metavar imply not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar imply not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma inSeries as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote metavar var sx end metavar end quote ) endorse metavar var sy end metavar in0 metavar var fx end metavar infer not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end define end math ] "

" [ math define proof of lemma inSeries as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var sx end metavar indeed all metavar var sy end metavar indeed lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote metavar var sx end metavar end quote ) endorse metavar var sy end metavar in0 metavar var fx end metavar infer axiom seriesType modus probans lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote metavar var sx end metavar end quote ) conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut 1rule repetition modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut prop lemma first conjunct modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut 1rule repetition modus ponens not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var conclude not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut prop lemma first conjunct modus ponens not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var conclude for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule repetition modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma a4 at metavar var sy end metavar modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude metavar var sy end metavar in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule mp modus ponens metavar var sy end metavar in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens metavar var sy end metavar in0 metavar var fx end metavar conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut 1rule repetition modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma inSeries helper conclude metavar var sy end metavar in0 metavar var fx end metavar imply not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar imply not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair cut prop lemma mp2 modus ponens metavar var sy end metavar in0 metavar var fx end metavar imply not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar imply not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair modus ponens metavar var sy end metavar in0 metavar var fx end metavar modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair cut pred lemma 2exist mp modus ponens not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair modus ponens not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 metavar var sx end metavar imply not0 metavar var sy end metavar = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair conclude not0 for all objects metavar var m end metavar indeed not0 metavar var sy end metavar = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma to=f subset helper as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar imply for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var s1 end var in0 metavar var fy end metavar end define end math ] "

" [ math define proof of lemma to=f subset helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar infer for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer lemma eqReflexivity conclude metavar var m end metavar = metavar var m end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma sameOrderedPair modus ponens metavar var m end metavar = metavar var m end metavar modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair cut lemma eqTransitivity modus ponens object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair conclude object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair cut lemma eqSymmetry modus ponens object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair = object var var s1 end var cut axiom natType conclude metavar var m end metavar in0 N cut lemma memberOfSeries modus ponens metavar var m end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair in0 metavar var fy end metavar cut lemma sameMember modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair = object var var s1 end var modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair in0 metavar var fy end metavar conclude object var var s1 end var in0 metavar var fy end metavar cut all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar infer for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer object var var s1 end var in0 metavar var fy end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar imply for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var s1 end var in0 metavar var fy end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma to=f subset as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) endorse lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) endorse for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar end define end math ] "

" [ math define proof of lemma to=f subset as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar infer for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer object var var s1 end var in0 metavar var fx end metavar infer lemma inSeries conclude not0 for all objects metavar var m end metavar indeed not0 object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair cut lemma to=f subset helper conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar imply for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var s1 end var in0 metavar var fy end metavar cut prop lemma mp2 modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar imply for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var s1 end var in0 metavar var fy end metavar modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar modus ponens for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var s1 end var in0 metavar var fy end metavar cut pred lemma exist mp modus ponens object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var s1 end var in0 metavar var fy end metavar modus ponens not0 for all objects metavar var m end metavar indeed not0 object var var s1 end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude object var var s1 end var in0 metavar var fy end metavar cut all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar infer not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar infer for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer object var var s1 end var in0 metavar var fx end metavar infer object var var s1 end var in0 metavar var fy end metavar conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar imply not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar imply for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar cut lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) endorse lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) endorse for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer axiom seriesType modus probans lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut axiom seriesType modus probans lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar cut prop lemma mp3 modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar imply not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar imply for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fy end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fy end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fy end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fy end metavar modus ponens for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar cut 1rule gen modus ponens object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar cut 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma to=f as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) endorse lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) endorse for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer metavar var fx end metavar = metavar var fy end metavar end define end math ] "

" [ math define proof of lemma to=f as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) endorse lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) endorse for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut 1rule gen modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude for all objects metavar var m end metavar indeed [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma to=f subset modus probans lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) modus probans lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) modus ponens for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar cut lemma to=f subset modus probans lambda var c dot typeSeries0( quote metavar var fy end metavar end quote , quote Q end quote ) modus probans lambda var c dot typeSeries0( quote metavar var fx end metavar end quote , quote Q end quote ) modus ponens for all objects metavar var m end metavar indeed [ metavar var fy end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fy end metavar imply object var var s1 end var in0 metavar var fx end metavar cut lemma set equality suff condition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fx end metavar imply object var var s1 end var in0 metavar var fy end metavar modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var fy end metavar imply object var var s1 end var in0 metavar var fx end metavar conclude metavar var fx end metavar = metavar var fy end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of tester1 as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer metavar var fx end metavar = metavar var fy end metavar end define end math ] "

" [ math define proof of tester1 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer lemma to=f modus ponens for all objects metavar var m end metavar indeed [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude metavar var fx end metavar = metavar var fy end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma reciprocalToLeft(Less) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar * 1/ metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar * 1/ metavar var z end metavar infer not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma reciprocalToLeft(Less) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var y end metavar * 1/ metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar * 1/ metavar var z end metavar infer lemma lessMultiplication modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar * 1/ metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar * 1/ metavar var z end metavar conclude not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * 1/ metavar var z end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * 1/ metavar var z end metavar * metavar var z end metavar cut lemma three2threeFactors conclude metavar var y end metavar * 1/ metavar var z end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar cut lemma positiveNonzero modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar conclude not0 metavar var z end metavar = 0 cut lemma x=x*y*(1/y) modus ponens not0 metavar var z end metavar = 0 conclude metavar var y end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar conclude metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar * 1/ metavar var z end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar * 1/ metavar var z end metavar = metavar var y end metavar conclude metavar var y end metavar * 1/ metavar var z end metavar * metavar var z end metavar = metavar var y end metavar cut lemma subLessRight modus ponens metavar var y end metavar * 1/ metavar var z end metavar * metavar var z end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * 1/ metavar var z end metavar * metavar var z end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * 1/ metavar var z end metavar * metavar var z end metavar conclude not0 metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar * metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "





--------------------

" [ math define statement of lemma toNumericalLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 - metavar var y end metavar <= metavar var x end metavar imply not0 not0 - metavar var y end metavar = metavar var x end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar end define end math ] "

" [ math define proof of lemma toNumericalLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 0 <= metavar var x end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma eqSymmetry modus ponens | metavar var x end metavar | = metavar var x end metavar conclude metavar var x end metavar = | metavar var x end metavar | cut lemma subLessLeft modus ponens metavar var x end metavar = | metavar var x end metavar | modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 - metavar var y end metavar <= metavar var x end metavar imply not0 not0 - metavar var y end metavar = metavar var x end metavar infer metavar var x end metavar <= 0 infer lemma lessNegated modus ponens not0 - metavar var y end metavar <= metavar var x end metavar imply not0 not0 - metavar var y end metavar = metavar var x end metavar conclude not0 - metavar var x end metavar <= - - metavar var y end metavar imply not0 not0 - metavar var x end metavar = - - metavar var y end metavar cut lemma nonpositiveNumerical modus ponens metavar var x end metavar <= 0 conclude | metavar var x end metavar | = - metavar var x end metavar cut lemma eqSymmetry modus ponens | metavar var x end metavar | = - metavar var x end metavar conclude - metavar var x end metavar = | metavar var x end metavar | cut lemma subLessLeft modus ponens - metavar var x end metavar = | metavar var x end metavar | modus ponens not0 - metavar var x end metavar <= - - metavar var y end metavar imply not0 not0 - metavar var x end metavar = - - metavar var y end metavar conclude not0 | metavar var x end metavar | <= - - metavar var y end metavar imply not0 not0 | metavar var x end metavar | = - - metavar var y end metavar cut lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma subLessRight modus ponens - - metavar var y end metavar = metavar var y end metavar modus ponens not0 | metavar var x end metavar | <= - - metavar var y end metavar imply not0 not0 | metavar var x end metavar | = - - metavar var y end metavar conclude not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 0 <= metavar var x end metavar infer not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 - metavar var y end metavar <= metavar var x end metavar imply not0 not0 - metavar var y end metavar = metavar var x end metavar infer metavar var x end metavar <= 0 infer not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar conclude not0 - metavar var y end metavar <= metavar var x end metavar imply not0 not0 - metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar <= 0 imply not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar cut not0 - metavar var y end metavar <= metavar var x end metavar imply not0 not0 - metavar var y end metavar = metavar var x end metavar infer not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar infer 1rule mp modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar imply 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar cut 1rule mp modus ponens not0 - metavar var y end metavar <= metavar var x end metavar imply not0 not0 - metavar var y end metavar = metavar var x end metavar imply metavar var x end metavar <= 0 imply not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar modus ponens not0 - metavar var y end metavar <= metavar var x end metavar imply not0 not0 - metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar <= 0 imply not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar modus ponens metavar var x end metavar <= 0 imply not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar conclude not0 | metavar var x end metavar | <= metavar var y end metavar imply not0 not0 | metavar var x end metavar | = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x<=|x| as system Q infer all metavar var x end metavar indeed metavar var x end metavar <= | metavar var x end metavar | end define end math ] "

" [ math define proof of lemma x<=|x| as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer lemma nonnegativeNumerical conclude | metavar var x end metavar | = metavar var x end metavar cut lemma eqSymmetry modus ponens | metavar var x end metavar | = metavar var x end metavar conclude metavar var x end metavar = | metavar var x end metavar | cut lemma eqLeq modus ponens metavar var x end metavar = | metavar var x end metavar | conclude metavar var x end metavar <= | metavar var x end metavar | cut all metavar var x end metavar indeed metavar var x end metavar <= 0 infer lemma 0<=|x| conclude 0 <= | metavar var x end metavar | cut lemma leqTransitivity modus ponens metavar var x end metavar <= 0 modus ponens 0 <= | metavar var x end metavar | conclude metavar var x end metavar <= | metavar var x end metavar | cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed 0 <= metavar var x end metavar infer metavar var x end metavar <= | metavar var x end metavar | conclude 0 <= metavar var x end metavar imply metavar var x end metavar <= | metavar var x end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed metavar var x end metavar <= 0 infer metavar var x end metavar <= | metavar var x end metavar | conclude metavar var x end metavar <= 0 imply metavar var x end metavar <= | metavar var x end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply metavar var x end metavar <= | metavar var x end metavar | modus ponens metavar var x end metavar <= 0 imply metavar var x end metavar <= | metavar var x end metavar | conclude metavar var x end metavar <= | metavar var x end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveToLeft(Less) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar infer not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma positiveToLeft(Less) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar infer lemma lessAddition modus ponens not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma subLessRight modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma negativeToRight(Less) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma negativeToRight(Less) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar infer lemma lessAddition modus ponens not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar conclude not0 metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var z end metavar + metavar var y end metavar cut lemma three2threeTerms conclude metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar conclude metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar cut lemma subLessLeft modus ponens metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var x end metavar modus ponens not0 metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar + metavar var y end metavar = metavar var z end metavar + metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma numericalDifferenceLess helper as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var x end metavar + - metavar var y end metavar infer not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar infer not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma numericalDifferenceLess helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var x end metavar + - metavar var y end metavar infer not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar infer lemma leqLessTransitivity modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar modus ponens not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar conclude not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar cut lemma positiveNegated modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar conclude not0 - metavar var z end metavar <= 0 imply not0 not0 - metavar var z end metavar = 0 cut lemma lessAdditionLeft modus ponens not0 - metavar var z end metavar <= 0 imply not0 not0 - metavar var z end metavar = 0 conclude not0 metavar var y end metavar + - metavar var z end metavar <= metavar var y end metavar + 0 imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var y end metavar + 0 cut axiom plus0 conclude metavar var y end metavar + 0 = metavar var y end metavar cut lemma subLessRight modus ponens metavar var y end metavar + 0 = metavar var y end metavar modus ponens not0 metavar var y end metavar + - metavar var z end metavar <= metavar var y end metavar + 0 imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var y end metavar + 0 conclude not0 metavar var y end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma negativeToLeft(Leq)(1 term) modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma lessLeqTransitivity modus ponens not0 metavar var y end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar cut lemma negativeToRight(Less) modus ponens not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar + metavar var y end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar cut lemma subLessRight modus ponens metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar + metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma join conjuncts modus ponens not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma numericalDifferenceLess as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar end define end math ] "

" [ math define proof of lemma numericalDifferenceLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer 0 <= metavar var x end metavar + - metavar var y end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar conclude | metavar var x end metavar + - metavar var y end metavar | = metavar var x end metavar + - metavar var y end metavar cut lemma subLessLeft modus ponens | metavar var x end metavar + - metavar var y end metavar | = metavar var x end metavar + - metavar var y end metavar modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar conclude not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar cut lemma numericalDifferenceLess helper modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar modus ponens not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar conclude not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer not0 0 <= metavar var x end metavar + - metavar var y end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar + - metavar var y end metavar conclude not0 metavar var x end metavar + - metavar var y end metavar <= 0 imply not0 not0 metavar var x end metavar + - metavar var y end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var x end metavar + - metavar var y end metavar <= 0 imply not0 not0 metavar var x end metavar + - metavar var y end metavar = 0 conclude | metavar var x end metavar + - metavar var y end metavar | = - metavar var x end metavar + - metavar var y end metavar cut lemma minusNegated conclude - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar cut lemma eqTransitivity modus ponens | metavar var x end metavar + - metavar var y end metavar | = - metavar var x end metavar + - metavar var y end metavar modus ponens - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar conclude | metavar var x end metavar + - metavar var y end metavar | = metavar var y end metavar + - metavar var x end metavar cut lemma subLessLeft modus ponens | metavar var x end metavar + - metavar var y end metavar | = metavar var y end metavar + - metavar var x end metavar modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar conclude not0 metavar var y end metavar + - metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar + - metavar var x end metavar = metavar var z end metavar cut lemma negativeNegated modus ponens not0 metavar var x end metavar + - metavar var y end metavar <= 0 imply not0 not0 metavar var x end metavar + - metavar var y end metavar = 0 conclude not0 0 <= - metavar var x end metavar + - metavar var y end metavar imply not0 not0 0 = - metavar var x end metavar + - metavar var y end metavar cut lemma subLessRight modus ponens - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar modus ponens not0 0 <= - metavar var x end metavar + - metavar var y end metavar imply not0 not0 0 = - metavar var x end metavar + - metavar var y end metavar conclude not0 0 <= metavar var y end metavar + - metavar var x end metavar imply not0 not0 0 = metavar var y end metavar + - metavar var x end metavar cut lemma lessLeq modus ponens not0 0 <= metavar var y end metavar + - metavar var x end metavar imply not0 not0 0 = metavar var y end metavar + - metavar var x end metavar conclude 0 <= metavar var y end metavar + - metavar var x end metavar cut lemma numericalDifferenceLess helper modus ponens 0 <= metavar var y end metavar + - metavar var x end metavar modus ponens not0 metavar var y end metavar + - metavar var x end metavar <= metavar var z end metavar imply not0 not0 metavar var y end metavar + - metavar var x end metavar = metavar var z end metavar conclude not0 not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar imply not0 not0 metavar var y end metavar <= metavar var x end metavar + metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + metavar var z end metavar cut prop lemma first conjunct modus ponens not0 not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar imply not0 not0 metavar var y end metavar <= metavar var x end metavar + metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + metavar var z end metavar conclude not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma negativeToRight(Less) modus ponens not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma second conjunct modus ponens not0 not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar imply not0 not0 metavar var y end metavar <= metavar var x end metavar + metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + metavar var z end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar + metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + metavar var z end metavar cut lemma positiveToLeft(Less) modus ponens not0 metavar var y end metavar <= metavar var x end metavar + metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + metavar var z end metavar conclude not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar cut prop lemma join conjuncts modus ponens not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer 0 <= metavar var x end metavar + - metavar var y end metavar infer not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar imply 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer not0 0 <= metavar var x end metavar + - metavar var y end metavar infer not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar imply not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer 1rule mp modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar imply 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar conclude 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut 1rule mp modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar imply not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar conclude not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma from negations modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma fpart-Bounded base as system Q infer all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var fx end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= 0 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar end define end math ] "

" [ math define proof of lemma fpart-Bounded base as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar <= 0 infer lemma leqLessEq modus ponens metavar var v2n end metavar <= 0 conclude not0 not0 metavar var v2n end metavar <= 0 imply not0 not0 metavar var v2n end metavar = 0 imply metavar var v2n end metavar = 0 cut axiom nonnegative(N) conclude 0 <= metavar var v2n end metavar cut lemma toNotLess modus ponens 0 <= metavar var v2n end metavar conclude not0 not0 metavar var v2n end metavar <= 0 imply not0 not0 metavar var v2n end metavar = 0 cut prop lemma negate first disjunct modus ponens not0 not0 metavar var v2n end metavar <= 0 imply not0 not0 metavar var v2n end metavar = 0 imply metavar var v2n end metavar = 0 modus ponens not0 not0 metavar var v2n end metavar <= 0 imply not0 not0 metavar var v2n end metavar = 0 conclude metavar var v2n end metavar = 0 cut lemma sameSeries modus ponens metavar var v2n end metavar = 0 conclude [ metavar var fx end metavar ; metavar var v2n end metavar ] = [ metavar var fx end metavar ; 0 ] cut lemma sameNumerical modus ponens [ metavar var fx end metavar ; metavar var v2n end metavar ] = [ metavar var fx end metavar ; 0 ] conclude | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; 0 ] | cut lemma eqAddition modus ponens | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; 0 ] | conclude | [ metavar var fx end metavar ; metavar var v2n end metavar ] | + 1 = | [ metavar var fx end metavar ; 0 ] | + 1 cut axiom leqReflexivity conclude | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; metavar var v2n end metavar ] | cut lemma leqPlus1 modus ponens | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; metavar var v2n end metavar ] | conclude not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; metavar var v2n end metavar ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; metavar var v2n end metavar ] | + 1 cut lemma subLessRight modus ponens | [ metavar var fx end metavar ; metavar var v2n end metavar ] | + 1 = | [ metavar var fx end metavar ; 0 ] | + 1 modus ponens not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; metavar var v2n end metavar ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; metavar var v2n end metavar ] | + 1 conclude not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; 0 ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; 0 ] | + 1 cut all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar <= 0 infer not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; 0 ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; 0 ] | + 1 conclude metavar var v2n end metavar <= 0 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; 0 ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; 0 ] | + 1 cut 1rule gen modus ponens metavar var v2n end metavar <= 0 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; 0 ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; 0 ] | + 1 conclude for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= 0 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; 0 ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; 0 ] | + 1 cut pred lemma intro exist at | [ metavar var fx end metavar ; 0 ] | + 1 modus ponens for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= 0 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; 0 ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; 0 ] | + 1 conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= 0 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma fpart-Bounded indu helper as system Q infer all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar infer metavar var v2n end metavar <= metavar var n end metavar + 1 infer not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) end define end math ] "

" [ math define proof of lemma fpart-Bounded indu helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar infer metavar var v2n end metavar <= metavar var n end metavar + 1 infer lemma leqLessEq modus ponens metavar var v2n end metavar <= metavar var n end metavar + 1 conclude not0 not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 imply metavar var v2n end metavar = metavar var n end metavar + 1 cut all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar infer not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 infer 1rule lessMinus1(N) modus ponens not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 conclude metavar var v2n end metavar <= metavar var n end metavar cut 1rule mp modus ponens metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar modus ponens metavar var v2n end metavar <= metavar var n end metavar conclude not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar cut lemma leqMax1 conclude metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut lemma lessLeqTransitivity modus ponens not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar modus ponens metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) conclude not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar = metavar var n end metavar + 1 infer lemma sameSeries modus ponens metavar var v2n end metavar = metavar var n end metavar + 1 conclude [ metavar var fx end metavar ; metavar var v2n end metavar ] = [ metavar var fx end metavar ; metavar var n end metavar + 1 ] cut lemma sameNumerical modus ponens [ metavar var fx end metavar ; metavar var v2n end metavar ] = [ metavar var fx end metavar ; metavar var n end metavar + 1 ] conclude | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | cut lemma eqLeq modus ponens | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | conclude | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | cut lemma leqPlus1 modus ponens | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | conclude not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 cut lemma leqMax2 conclude | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut lemma lessLeqTransitivity modus ponens not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 modus ponens | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) conclude not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar infer not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 infer not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) conclude metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar imply not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut 1rule mp modus ponens metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar imply not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) modus ponens metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar conclude not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar = metavar var n end metavar + 1 infer not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) conclude metavar var v2n end metavar = metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut prop lemma from disjuncts modus ponens not0 not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 imply metavar var v2n end metavar = metavar var n end metavar + 1 modus ponens not0 metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 not0 metavar var v2n end metavar = metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) modus ponens metavar var v2n end metavar = metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) conclude not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fpart-Bounded indu as system Q infer all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar end define end math ] "


" [ math define proof of lemma fpart-Bounded indu as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar infer metavar var v2n end metavar <= metavar var n end metavar + 1 infer lemma fpart-Bounded indu helper modus ponens metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar modus ponens metavar var v2n end metavar <= metavar var n end metavar + 1 conclude not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2n end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar infer metavar var v2n end metavar <= metavar var n end metavar + 1 infer not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) conclude metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar imply metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut pred lemma addAll modus ponens metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar imply metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) conclude for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar imply for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) cut pred lemma addExist(SimpleAnt) at if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) modus ponens for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar imply for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 <= metavar var v1 end metavar , metavar var v1 end metavar , | [ metavar var fx end metavar ; metavar var n end metavar + 1 ] | + 1 ) conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2n end metavar indeed metavar var v2n end metavar <= metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2n end metavar ] | = metavar var v1 end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma fpart-Bounded as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar end define end math ] "

" [ math define proof of lemma fpart-Bounded as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed lemma fpart-Bounded base conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= 0 imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut lemma fpart-Bounded indu conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut lemma induction modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= 0 imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar + 1 imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma f-Bounded helper as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer not0 | metavar var y end metavar | <= if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) imply not0 not0 | metavar var y end metavar | = if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) end define end math ] "

" [ math define proof of lemma f-Bounded helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar infer lemma numericalDifferenceLess modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = metavar var z end metavar conclude not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma first conjunct modus ponens not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar cut lemma negativeToRight(Less) modus ponens not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar conclude not0 metavar var y end metavar <= metavar var x end metavar + metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + metavar var z end metavar cut lemma x<=|x| conclude metavar var x end metavar + metavar var z end metavar <= | metavar var x end metavar + metavar var z end metavar | cut lemma lessLeqTransitivity modus ponens not0 metavar var y end metavar <= metavar var x end metavar + metavar var z end metavar imply not0 not0 metavar var y end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var x end metavar + metavar var z end metavar <= | metavar var x end metavar + metavar var z end metavar | conclude not0 metavar var y end metavar <= | metavar var x end metavar + metavar var z end metavar | imply not0 not0 metavar var y end metavar = | metavar var x end metavar + metavar var z end metavar | cut lemma leqMax1 conclude | metavar var x end metavar + metavar var z end metavar | <= if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) cut lemma lessLeqTransitivity modus ponens not0 metavar var y end metavar <= | metavar var x end metavar + metavar var z end metavar | imply not0 not0 metavar var y end metavar = | metavar var x end metavar + metavar var z end metavar | modus ponens | metavar var x end metavar + metavar var z end metavar | <= if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) conclude not0 metavar var y end metavar <= if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) imply not0 not0 metavar var y end metavar = if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) cut prop lemma second conjunct modus ponens not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut lemma positiveToLeft(Less) modus ponens not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma x<=|x| conclude metavar var z end metavar + - metavar var x end metavar <= | metavar var z end metavar + - metavar var x end metavar | cut lemma numericalDifference conclude | metavar var z end metavar + - metavar var x end metavar | = | metavar var x end metavar + - metavar var z end metavar | cut lemma subLeqRight modus ponens | metavar var z end metavar + - metavar var x end metavar | = | metavar var x end metavar + - metavar var z end metavar | modus ponens metavar var z end metavar + - metavar var x end metavar <= | metavar var z end metavar + - metavar var x end metavar | conclude metavar var z end metavar + - metavar var x end metavar <= | metavar var x end metavar + - metavar var z end metavar | cut lemma leqNegated modus ponens metavar var z end metavar + - metavar var x end metavar <= | metavar var x end metavar + - metavar var z end metavar | conclude - | metavar var x end metavar + - metavar var z end metavar | <= - metavar var z end metavar + - metavar var x end metavar cut lemma minusNegated conclude - metavar var z end metavar + - metavar var x end metavar = metavar var x end metavar + - metavar var z end metavar cut lemma subLeqRight modus ponens - metavar var z end metavar + - metavar var x end metavar = metavar var x end metavar + - metavar var z end metavar modus ponens - | metavar var x end metavar + - metavar var z end metavar | <= - metavar var z end metavar + - metavar var x end metavar conclude - | metavar var x end metavar + - metavar var z end metavar | <= metavar var x end metavar + - metavar var z end metavar cut lemma leqLessTransitivity modus ponens - | metavar var x end metavar + - metavar var z end metavar | <= metavar var x end metavar + - metavar var z end metavar modus ponens not0 metavar var x end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + - metavar var z end metavar = metavar var y end metavar conclude not0 - | metavar var x end metavar + - metavar var z end metavar | <= metavar var y end metavar imply not0 not0 - | metavar var x end metavar + - metavar var z end metavar | = metavar var y end metavar cut lemma leqMax2 conclude | metavar var x end metavar + - metavar var z end metavar | <= if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) cut lemma leqNegated modus ponens | metavar var x end metavar + - metavar var z end metavar | <= if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) conclude - if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) <= - | metavar var x end metavar + - metavar var z end metavar | cut lemma leqLessTransitivity modus ponens - if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) <= - | metavar var x end metavar + - metavar var z end metavar | modus ponens not0 - | metavar var x end metavar + - metavar var z end metavar | <= metavar var y end metavar imply not0 not0 - | metavar var x end metavar + - metavar var z end metavar | = metavar var y end metavar conclude not0 - if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) <= metavar var y end metavar imply not0 not0 - if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) = metavar var y end metavar cut lemma toNumericalLess modus ponens not0 - if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) <= metavar var y end metavar imply not0 not0 - if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) = metavar var y end metavar modus ponens not0 metavar var y end metavar <= if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) imply not0 not0 metavar var y end metavar = if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) conclude not0 | metavar var y end metavar | <= if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) imply not0 not0 | metavar var y end metavar | = if( | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + metavar var z end metavar | , | metavar var x end metavar + - metavar var z end metavar | ) end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma f-Bounded as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar end define end math ] "

" [ math define proof of lemma f-Bounded as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 infer metavar var n end metavar <= metavar var v2 end metavar infer lemma a4 at metavar var n end metavar modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 conclude for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var n end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 cut lemma a4 at metavar var v2 end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var n end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 conclude not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var n end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 cut lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut axiom leqReflexivity conclude metavar var n end metavar <= metavar var n end metavar cut prop lemma mp3 modus ponens not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var n end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 modus ponens not0 0 <= 1 imply not0 not0 0 = 1 modus ponens metavar var n end metavar <= metavar var n end metavar modus ponens metavar var n end metavar <= metavar var v2 end metavar conclude not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 cut lemma f-Bounded helper modus ponens not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 conclude not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar infer not0 metavar var n end metavar <= metavar var v2 end metavar infer lemma a4 at metavar var v2 end metavar modus ponens for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut lemma toLess modus ponens not0 metavar var n end metavar <= metavar var v2 end metavar conclude not0 metavar var v2 end metavar <= metavar var n end metavar imply not0 not0 metavar var v2 end metavar = metavar var n end metavar cut lemma lessLeq modus ponens not0 metavar var v2 end metavar <= metavar var n end metavar imply not0 not0 metavar var v2 end metavar = metavar var n end metavar conclude metavar var v2 end metavar <= metavar var n end metavar cut 1rule mp modus ponens metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar modus ponens metavar var v2 end metavar <= metavar var n end metavar conclude not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 infer metavar var n end metavar <= metavar var v2 end metavar infer not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) conclude for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) cut 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar infer not0 metavar var n end metavar <= metavar var v2 end metavar infer not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut axiom cauchy conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma a4 at 1 modus ponens for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 cut lemma fpart-Bounded conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut pred lemma exist mp modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1 imply not0 not0 0 = 1 imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1 imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1 conclude metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) cut pred lemma exist mp modus ponens for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed metavar var v2 end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude not0 metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut lemma lessThanMax modus ponens metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) modus ponens not0 metavar var n end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) cut 1rule gen modus ponens not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) conclude for all objects metavar var v2 end metavar indeed not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) cut pred lemma intro exist at if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) modus ponens for all objects metavar var v2 end metavar indeed not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = if( metavar var v1 end metavar <= if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , if( | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | <= | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + 1 | , | [ metavar var fx end metavar ; metavar var n end metavar ] + - 1 | ) , metavar var v1 end metavar ) conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar end quote state proof state cache var c end expand end define end math ] "






" [ math define statement of lemma negativeToLeft(Less) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar infer not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma negativeToLeft(Less) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar infer lemma lessAddition modus ponens not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + - metavar var z end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar cut lemma three2threeTerms conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma x=x+y-y conclude metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar conclude metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar modus ponens metavar var y end metavar + metavar var z end metavar + - metavar var z end metavar = metavar var y end metavar conclude metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar cut lemma subLessRight modus ponens metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + - metavar var z end metavar + metavar var z end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma positiveTripled as system Q infer all metavar var x end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var x end metavar infer not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar end define end math ] "

" [ math define proof of lemma positiveTripled as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var x end metavar infer lemma 0<3 conclude not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 cut lemma positiveFactors modus ponens not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 modus ponens not0 0 <= 1/ 1 + 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var x end metavar conclude not0 0 <= 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut axiom timesAssociativity conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma positiveNonzero modus ponens not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 conclude not0 1 + 1 + 1 = 0 cut lemma reciprocal modus ponens not0 1 + 1 + 1 = 0 conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 = 1 cut lemma eqMultiplication modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 = 1 conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 * metavar var x end metavar cut lemma times1Left conclude 1 * metavar var x end metavar = metavar var x end metavar cut lemma eqTransitivity4 modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 * metavar var x end metavar modus ponens 1 * metavar var x end metavar = metavar var x end metavar conclude 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar cut lemma subLessRight modus ponens 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar modus ponens not0 0 <= 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1 + 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveDividedBy3 as system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer not0 0 <= 1/ 1 + 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var x end metavar end define end math ] "

" [ math define proof of lemma positiveDividedBy3 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma 0<3 conclude not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 cut lemma positiveInverted modus ponens not0 0 <= 1 + 1 + 1 imply not0 not0 0 = 1 + 1 + 1 conclude not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 cut lemma positiveFactors modus ponens not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 <= 1/ 1 + 1 + 1 * metavar var x end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma |x-x|=0 as system Q infer all metavar var x end metavar indeed | metavar var x end metavar + - metavar var x end metavar | = 0 end define end math ] "

" [ math define proof of lemma |x-x|=0 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma eqReflexivity conclude metavar var x end metavar = metavar var x end metavar cut lemma positiveToLeft(Eq)(1 term) modus ponens metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar + - metavar var x end metavar = 0 cut lemma sameNumerical modus ponens metavar var x end metavar + - metavar var x end metavar = 0 conclude | metavar var x end metavar + - metavar var x end metavar | = | 0 | cut lemma |0|=0 conclude | 0 | = 0 cut lemma eqTransitivity modus ponens | metavar var x end metavar + - metavar var x end metavar | = | 0 | modus ponens | 0 | = 0 conclude | metavar var x end metavar + - metavar var x end metavar | = 0 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma 1<2 as system Q infer not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 end define end math ] "

" [ math define proof of lemma 1<2 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma lessAddition modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 0 + 1 <= 1 + 1 imply not0 not0 0 + 1 = 1 + 1 cut lemma plus0Left conclude 0 + 1 = 1 cut lemma subLessLeft modus ponens 0 + 1 = 1 modus ponens not0 0 + 1 <= 1 + 1 imply not0 not0 0 + 1 = 1 + 1 conclude not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 cut 1rule repetition modus ponens not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 conclude not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma 1/3<2/3 as system Q infer not0 1/ 1 + 1 + 1 <= 1 + 1 * 1/ 1 + 1 + 1 imply not0 not0 1/ 1 + 1 + 1 = 1 + 1 * 1/ 1 + 1 + 1 end define end math ] "

" [ math define proof of lemma 1/3<2/3 as lambda var c dot lambda var x dot proof expand quote system Q infer lemma 1<2 conclude not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 cut lemma 0<1/3 conclude not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 cut lemma lessMultiplication modus ponens not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 modus ponens not0 1 <= 1 + 1 imply not0 not0 1 = 1 + 1 conclude not0 1 * 1/ 1 + 1 + 1 <= 1 + 1 * 1/ 1 + 1 + 1 imply not0 not0 1 * 1/ 1 + 1 + 1 = 1 + 1 * 1/ 1 + 1 + 1 cut lemma times1Left conclude 1 * 1/ 1 + 1 + 1 = 1/ 1 + 1 + 1 cut lemma subLessLeft modus ponens 1 * 1/ 1 + 1 + 1 = 1/ 1 + 1 + 1 modus ponens not0 1 * 1/ 1 + 1 + 1 <= 1 + 1 * 1/ 1 + 1 + 1 imply not0 not0 1 * 1/ 1 + 1 + 1 = 1 + 1 * 1/ 1 + 1 + 1 conclude not0 1/ 1 + 1 + 1 <= 1 + 1 * 1/ 1 + 1 + 1 imply not0 not0 1/ 1 + 1 + 1 = 1 + 1 * 1/ 1 + 1 + 1 cut 1rule repetition modus ponens not0 1/ 1 + 1 + 1 <= 1 + 1 * 1/ 1 + 1 + 1 imply not0 not0 1/ 1 + 1 + 1 = 1 + 1 * 1/ 1 + 1 + 1 conclude not0 1/ 1 + 1 + 1 <= 1 + 1 * 1/ 1 + 1 + 1 imply not0 not0 1/ 1 + 1 + 1 = 1 + 1 * 1/ 1 + 1 + 1 end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma (1/3)x+(1/3)x=(2/3)x as system Q infer all metavar var x end metavar indeed 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar end define end math ] "

" [ math define proof of lemma (1/3)x+(1/3)x=(2/3)x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma x+x=2*x conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut axiom timesAssociativity conclude 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqTransitivity modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar modus ponens 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut 1rule repetition modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma -(1/3)x-(1/3)x=-(2/3)x as system Q infer all metavar var x end metavar indeed - 1/ 1 + 1 + 1 * metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar end define end math ] "

" [ math define proof of lemma -(1/3)x-(1/3)x=-(2/3)x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma (1/3)x+(1/3)x=(2/3)x conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqNegated modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude - 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma -x-y=-(x+y) conclude - 1/ 1 + 1 + 1 * metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar = - 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqTransitivity modus ponens - 1/ 1 + 1 + 1 * metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar = - 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar modus ponens - 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude - 1/ 1 + 1 + 1 * metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma (2/3)x+(1/3)x=x as system Q infer all metavar var x end metavar indeed 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma (2/3)x+(1/3)x=x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma x+x=2*x conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut axiom timesAssociativity conclude 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqTransitivity modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar modus ponens 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqAddition modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma (1/3)x+(1/3)x+(1/3)x=x conclude 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar cut lemma equality modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar conclude 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma -x+(1/3)x=-(2/3)x as system Q infer all metavar var x end metavar indeed - metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar end define end math ] "

" [ math define proof of lemma -x+(1/3)x=-(2/3)x as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed lemma (2/3)x+(1/3)x=x conclude 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar cut lemma positiveToRight(Eq) modus ponens 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar conclude 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqNegated modus ponens 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar conclude - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = - metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma minusNegated conclude - metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar = 1/ 1 + 1 + 1 * metavar var x end metavar + - metavar var x end metavar cut axiom plusCommutativity conclude 1/ 1 + 1 + 1 * metavar var x end metavar + - metavar var x end metavar = - metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqTransitivity4 modus ponens - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = - metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar modus ponens - metavar var x end metavar + - 1/ 1 + 1 + 1 * metavar var x end metavar = 1/ 1 + 1 + 1 * metavar var x end metavar + - metavar var x end metavar modus ponens 1/ 1 + 1 + 1 * metavar var x end metavar + - metavar var x end metavar = - metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar conclude - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = - metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar cut lemma eqSymmetry modus ponens - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar = - metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar conclude - metavar var x end metavar + 1/ 1 + 1 + 1 * metavar var x end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var x end metavar end quote state proof state cache var c end expand end define 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 define statement of lemma preserveLessGreater as system Q infer all metavar var x1 end metavar indeed all metavar var x2 end metavar indeed all metavar var y1 end metavar indeed all metavar var y2 end metavar indeed all metavar var z end metavar indeed metavar var x1 end metavar <= metavar var y1 end metavar + - metavar var z end metavar infer not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer metavar var x2 end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar end define end math ] "

" [ math define proof of lemma preserveLessGreater as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x1 end metavar indeed all metavar var x2 end metavar indeed all metavar var y1 end metavar indeed all metavar var y2 end metavar indeed all metavar var z end metavar indeed metavar var x1 end metavar <= metavar var y1 end metavar + - metavar var z end metavar infer not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer lemma 0<=|x| conclude 0 <= | metavar var x1 end metavar + - metavar var x2 end metavar | cut lemma leqLessTransitivity modus ponens 0 <= | metavar var x1 end metavar + - metavar var x2 end metavar | modus ponens not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 0 <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma positiveTripled modus ponens not0 0 <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar cut lemma numericalDifferenceLess modus ponens not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var x1 end metavar imply not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var x1 end metavar imply not0 not0 metavar var x1 end metavar <= metavar var x2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x1 end metavar = metavar var x2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut prop lemma first conjunct modus ponens not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var x1 end metavar imply not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var x1 end metavar imply not0 not0 metavar var x1 end metavar <= metavar var x2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x1 end metavar = metavar var x2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var x1 end metavar imply not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var x1 end metavar cut lemma negativeToRight(Less) modus ponens not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var x1 end metavar imply not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var x1 end metavar conclude not0 metavar var x2 end metavar <= metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma leqAddition modus ponens metavar var x1 end metavar <= metavar var y1 end metavar + - metavar var z end metavar conclude metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar + - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma -x+(1/3)x=-(2/3)x conclude - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma three2twoTerms modus ponens - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var y1 end metavar + - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma subLeqRight modus ponens metavar var y1 end metavar + - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar + - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma numericalDifferenceLess modus ponens not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 not0 metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar imply not0 not0 metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar imply not0 not0 metavar var y1 end metavar <= metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar = metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut prop lemma second conjunct modus ponens not0 not0 metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar imply not0 not0 metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar imply not0 not0 metavar var y1 end metavar <= metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar = metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var y1 end metavar <= metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar = metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma positiveToLeft(Less) modus ponens not0 metavar var y1 end metavar <= metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar = metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar imply not0 not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar cut lemma lessAddition modus ponens not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar imply not0 not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar conclude not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar cut axiom plusAssociativity conclude metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma -(1/3)x-(1/3)x=-(2/3)x conclude - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma eqAdditionLeft modus ponens - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma subLessLeft modus ponens metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma lessLeqTransitivity modus ponens not0 metavar var x2 end metavar <= metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var x2 end metavar <= metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma lessTransitivity modus ponens not0 metavar var x2 end metavar <= metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens not0 metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var x2 end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma lessLeq modus ponens not0 metavar var x2 end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var x2 end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma closetolessIsLess as system Q infer all metavar var x1 end metavar indeed all metavar var x2 end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x1 end metavar <= metavar var y end metavar + - metavar var z end metavar infer not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer metavar var x2 end metavar <= metavar var y end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar end define end math ] "

" [ math define proof of lemma closetolessIsLess as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x1 end metavar indeed all metavar var x2 end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x1 end metavar <= metavar var y end metavar + - metavar var z end metavar infer not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer lemma 0<=|x| conclude 0 <= | metavar var x1 end metavar + - metavar var x2 end metavar | cut lemma leqLessTransitivity modus ponens 0 <= | metavar var x1 end metavar + - metavar var x2 end metavar | modus ponens not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 0 <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma |x-x|=0 conclude | metavar var y end metavar + - metavar var y end metavar | = 0 cut lemma eqSymmetry modus ponens | metavar var y end metavar + - metavar var y end metavar | = 0 conclude 0 = | metavar var y end metavar + - metavar var y end metavar | cut lemma subLessLeft modus ponens 0 = | metavar var y end metavar + - metavar var y end metavar | modus ponens not0 0 <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 | metavar var y end metavar + - metavar var y end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y end metavar + - metavar var y end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma preserveLessGreater modus ponens metavar var x1 end metavar <= metavar var y end metavar + - metavar var z end metavar modus ponens not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens not0 | metavar var y end metavar + - metavar var y end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y end metavar + - metavar var y end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var x2 end metavar <= metavar var y end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma subLessLeft(F) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma subLessLeft(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer 1rule repetition modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule deduction modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n1 end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut lemma a4 at 1/ 1 + 1 + 1 * object var var ep end var modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var n1 end var indeed not0 for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut 1rule deduction modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n2 end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut all metavar var fx end metavar indeed all metavar var fx end metavar indeed all metavar var fz end metavar indeed for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer lemma a4 at object var var m end var modus ponens for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma first conjunct modus ponens not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var cut lemma positiveDividedBy3 modus ponens not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var conclude not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var infer for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var infer lemma leqMax1 conclude object var var n1 end var <= if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) cut lemma leqTransitivity modus ponens object var var n1 end var <= if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) modus ponens if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var conclude object var var n1 end var <= object var var m end var cut lemma leqMax2 conclude object var var n2 end var <= if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) cut lemma leqTransitivity modus ponens object var var n2 end var <= if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) modus ponens if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var conclude object var var n2 end var <= object var var m end var cut lemma a4 at object var var m end var modus ponens for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma first conjunct modus ponens not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var cut prop lemma second conjunct modus ponens not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut 1rule mp modus ponens object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var modus ponens object var var n2 end var <= object var var m end var conclude [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut lemma a4 at object var var m end var modus ponens for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var conclude not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var cut lemma positiveDividedBy3 modus ponens not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var conclude not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var cut prop lemma mp2 modus ponens not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var modus ponens object var var n1 end var <= object var var m end var conclude not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var cut lemma closetolessIsLess modus ponens [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var modus ponens not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var conclude [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fx end metavar indeed all metavar var fz end metavar indeed for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var conclude for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma 2exist mp modus ponens for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n2 end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var infer for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var infer [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var imply for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma exist mp modus ponens for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var imply for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 for all objects object var var n1 end var indeed not0 for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var conclude for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma 2exist mp modus ponens for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n2 end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut prop lemma join conjuncts modus ponens not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var modus ponens if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut 1rule gen modus ponens not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude for all objects object var var m end var indeed not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma intro exist at if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) modus ponens for all objects object var var m end var indeed not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma intro exist at 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma subLessLeft(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma subLessLeft(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer 1rule from== modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut lemma subLessLeft(F) modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma closetogreaterIsGreater as system Q infer all metavar var x end metavar indeed all metavar var y1 end metavar indeed all metavar var y2 end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y1 end metavar + - metavar var z end metavar infer not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer metavar var x end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar end define end math ] "

" [ math define proof of lemma closetogreaterIsGreater as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y1 end metavar indeed all metavar var y2 end metavar indeed all metavar var z end metavar indeed metavar var x end metavar <= metavar var y1 end metavar + - metavar var z end metavar infer not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer lemma 0<=|x| conclude 0 <= | metavar var y1 end metavar + - metavar var y2 end metavar | cut lemma leqLessTransitivity modus ponens 0 <= | metavar var y1 end metavar + - metavar var y2 end metavar | modus ponens not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 0 <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma |x-x|=0 conclude | metavar var x end metavar + - metavar var x end metavar | = 0 cut lemma eqSymmetry modus ponens | metavar var x end metavar + - metavar var x end metavar | = 0 conclude 0 = | metavar var x end metavar + - metavar var x end metavar | cut lemma subLessLeft modus ponens 0 = | metavar var x end metavar + - metavar var x end metavar | modus ponens not0 0 <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 | metavar var x end metavar + - metavar var x end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var x end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma preserveLessGreater modus ponens metavar var x end metavar <= metavar var y1 end metavar + - metavar var z end metavar modus ponens not0 | metavar var x end metavar + - metavar var x end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x end metavar + - metavar var x end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var x end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma subLessRight(F) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma subLessRight(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer 1rule repetition modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule deduction modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n1 end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut lemma a4 at 1/ 1 + 1 + 1 * object var var ep end var modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var n1 end var indeed not0 for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut 1rule deduction modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n2 end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut all metavar var fx end metavar indeed all metavar var fx end metavar indeed all metavar var fz end metavar indeed for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer lemma a4 at object var var m end var modus ponens for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma first conjunct modus ponens not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var cut lemma positiveDividedBy3 modus ponens not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var conclude not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var infer for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var infer lemma leqMax1 conclude object var var n1 end var <= if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) cut lemma leqTransitivity modus ponens object var var n1 end var <= if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) modus ponens if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var conclude object var var n1 end var <= object var var m end var cut lemma leqMax2 conclude object var var n2 end var <= if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) cut lemma leqTransitivity modus ponens object var var n2 end var <= if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) modus ponens if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var conclude object var var n2 end var <= object var var m end var cut lemma a4 at object var var m end var modus ponens for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma first conjunct modus ponens not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var cut prop lemma second conjunct modus ponens not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut 1rule mp modus ponens object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var modus ponens object var var n2 end var <= object var var m end var conclude [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut lemma a4 at object var var m end var modus ponens for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var conclude not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var cut lemma positiveDividedBy3 modus ponens not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var conclude not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var cut prop lemma mp2 modus ponens not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var modus ponens object var var n1 end var <= object var var m end var conclude not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var cut lemma closetogreaterIsGreater modus ponens [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var modus ponens not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var conclude [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fx end metavar indeed all metavar var fz end metavar indeed for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var conclude for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma 2exist mp modus ponens for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n2 end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var infer for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var infer [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var imply for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma exist mp modus ponens for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var imply for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 for all objects object var var n1 end var indeed not0 for all objects object var var m end var indeed not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply object var var n1 end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = 1/ 1 + 1 + 1 * object var var ep end var conclude for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma 2exist mp modus ponens for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n2 end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n2 end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut prop lemma join conjuncts modus ponens not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var modus ponens if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut 1rule gen modus ponens not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude for all objects object var var m end var indeed not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma intro exist at if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) modus ponens for all objects object var var m end var indeed not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 if( object var var n2 end var <= object var var n1 end var , object var var n1 end var , object var var n2 end var ) <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var cut pred lemma intro exist at 1/ 1 + 1 + 1 * object var var ep end var modus ponens not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= 1/ 1 + 1 + 1 * object var var ep end var imply not0 not0 0 = 1/ 1 + 1 + 1 * object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - 1/ 1 + 1 + 1 * object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma subLessRight(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma subLessRight(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var infer 1rule from== modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut lemma subLessRight(F) modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fz end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma -x*y=-(x*y) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed - metavar var x end metavar * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma -x*y=-(x*y) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed lemma times(-1)Left conclude - 1 * metavar var x end metavar = - metavar var x end metavar cut lemma eqMultiplication modus ponens - 1 * metavar var x end metavar = - metavar var x end metavar conclude - 1 * metavar var x end metavar * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar cut lemma eqSymmetry modus ponens - 1 * metavar var x end metavar * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar conclude - metavar var x end metavar * metavar var y end metavar = - 1 * metavar var x end metavar * metavar var y end metavar cut axiom timesAssociativity conclude - 1 * metavar var x end metavar * metavar var y end metavar = - 1 * metavar var x end metavar * metavar var y end metavar cut lemma times(-1)Left conclude - 1 * metavar var x end metavar * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar cut lemma eqTransitivity4 modus ponens - metavar var x end metavar * metavar var y end metavar = - 1 * metavar var x end metavar * metavar var y end metavar modus ponens - 1 * metavar var x end metavar * metavar var y end metavar = - 1 * metavar var x end metavar * metavar var y end metavar modus ponens - 1 * metavar var x end metavar * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar conclude - metavar var x end metavar * metavar var y end metavar = - metavar var x end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqMultiplicationLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar * metavar var x end metavar <= metavar var z end metavar * metavar var y end metavar end define end math ] "

" [ math define proof of lemma leqMultiplicationLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar infer lemma leqMultiplication modus ponens 0 <= metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var x end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var x end metavar modus ponens metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar conclude metavar var z end metavar * metavar var x end metavar <= metavar var y end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar cut lemma subLeqRight modus ponens metavar var y end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar modus ponens metavar var z end metavar * metavar var x end metavar <= metavar var y end metavar * metavar var z end metavar conclude metavar var z end metavar * metavar var x end metavar <= metavar var z end metavar * metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameFmultiplication helper as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar imply for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar end define end math ] "


" [ math define proof of lemma sameFmultiplication helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar infer for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer metavar var n end metavar <= metavar var m end metavar infer lemma a4 at metavar var v2 end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut lemma 0<=|x| conclude 0 <= | [ metavar var fz end metavar ; metavar var v2 end metavar ] | cut lemma leqLessTransitivity modus ponens 0 <= | [ metavar var fz end metavar ; metavar var v2 end metavar ] | modus ponens not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude not0 0 <= metavar var v1 end metavar imply not0 not0 0 = metavar var v1 end metavar cut lemma positiveInverted modus ponens not0 0 <= metavar var v1 end metavar imply not0 not0 0 = metavar var v1 end metavar conclude not0 0 <= 1/ metavar var v1 end metavar imply not0 not0 0 = 1/ metavar var v1 end metavar cut lemma positiveFactors modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar modus ponens not0 0 <= 1/ metavar var v1 end metavar imply not0 not0 0 = 1/ metavar var v1 end metavar conclude not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar conclude not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar cut prop lemma mp2 modus ponens not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar modus ponens not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar modus ponens metavar var n end metavar <= metavar var m end metavar conclude not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar cut lemma reciprocalToLeft(Less) modus ponens not0 0 <= metavar var v1 end metavar imply not0 not0 0 = metavar var v1 end metavar modus ponens not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar conclude not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar = metavar var ep end metavar cut lemma timesF conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma timesF conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqNegated modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] conclude - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma -x*y=-(x*y) conclude - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] conclude - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] conclude - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma addEquations modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] cut lemma distributionOutLeft conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma sameNumerical modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] conclude | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = | [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut lemma splitNumericalProduct conclude | [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = | [ metavar var fz end metavar ; metavar var m end metavar ] | * | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut axiom timesCommutativity conclude | [ metavar var fz end metavar ; metavar var m end metavar ] | * | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | cut lemma eqTransitivity4 modus ponens | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = | [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens | [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = | [ metavar var fz end metavar ; metavar var m end metavar ] | * | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens | [ metavar var fz end metavar ; metavar var m end metavar ] | * | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | conclude | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | cut lemma eqSymmetry modus ponens | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | conclude | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | = | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | cut lemma a4 at [ metavar var fz end metavar ; metavar var m end metavar ] modus ponens for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude | [ metavar var fz end metavar ; metavar var m end metavar ] | <= metavar var v1 end metavar cut lemma 0<=|x| conclude 0 <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut lemma leqMultiplicationLeft modus ponens 0 <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens | [ metavar var fz end metavar ; metavar var m end metavar ] | <= metavar var v1 end metavar conclude | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar cut lemma leqLessTransitivity modus ponens | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar modus ponens not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * metavar var v1 end metavar = metavar var ep end metavar conclude not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut lemma subLessLeft modus ponens | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | = | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | modus ponens not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | * | [ metavar var fz end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar infer for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer metavar var n end metavar <= metavar var m end metavar infer not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar imply for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma sameFmultiplication as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | = object var var ep end var end define end math ] "

" [ math define proof of lemma sameFmultiplication as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer 1rule repetition modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule deduction modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut lemma a4 at metavar var ep end metavar * 1/ metavar var v1 end metavar modus ponens for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar cut lemma f-Bounded conclude not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar cut lemma sameFmultiplication helper conclude for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar imply for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar cut pred lemma exist mp2 modus ponens for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar imply for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 0 = metavar var ep end metavar * 1/ metavar var v1 end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar * 1/ metavar var v1 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar * 1/ metavar var v1 end metavar modus ponens not0 for all objects metavar var v1 end metavar indeed not0 for all objects metavar var v2 end metavar indeed not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | <= metavar var v1 end metavar imply not0 not0 | [ metavar var fz end metavar ; metavar var v2 end metavar ] | = metavar var v1 end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar cut 1rule gen modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar cut pred lemma intro exist at metavar var n end metavar modus ponens for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar cut 1rule gen modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar cut 1rule deduction modus ponens for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | = object var var ep end var cut 1rule repetition modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | = object var var ep end var end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma eqMultiplication(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "


" [ math define proof of lemma eqMultiplication(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule from== modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut lemma sameFmultiplication modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | = object var var ep end var cut 1rule to== modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] | = object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqReflexivity conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqReflexivity conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqTransitivity4 modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x*0=0(F) as system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set end define end math ] "

" [ math define proof of lemma x*0=0(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var fx end metavar indeed lemma timesF conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] cut axiom natType conclude metavar var m end metavar in0 N cut lemma 0f modus ponens metavar var m end metavar in0 N conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 cut lemma eqMultiplicationLeft modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * 0 cut lemma x*0=0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] * 0 = 0 cut lemma eqSymmetry modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 conclude 0 = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] cut lemma eqTransitivity5 modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * 0 modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * 0 = 0 modus ponens 0 = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] cut 1rule gen modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] conclude for all objects metavar var m end metavar indeed [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] cut lemma to=f modus ponens for all objects metavar var m end metavar indeed [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] conclude the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma x*0=0(R) as system Q infer all metavar var fx end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma x*0=0(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma x*0=0(F) conclude the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set cut lemma equalsSameF modus ponens the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set = the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | = object var var ep end var cut 1rule to== modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | = object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqReflexivity conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Transitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule repetition modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma distributionLeft as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var x end metavar * metavar var y end metavar + metavar var z end metavar = metavar var y end metavar * metavar var x end metavar + metavar var z end metavar * metavar var x end metavar end define end math ] "

" [ math define proof of lemma distributionLeft as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma distributionOutLeft conclude metavar var y end metavar * metavar var x end metavar + metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar cut lemma eqSymmetry modus ponens metavar var y end metavar * metavar var x end metavar + metavar var z end metavar * metavar var x end metavar = metavar var x end metavar * metavar var y end metavar + metavar var z end metavar conclude metavar var x end metavar * metavar var y end metavar + metavar var z end metavar = metavar var y end metavar * metavar var x end metavar + metavar var z end metavar * metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

XX 'nonnegativeFactors' er en konsekvens heraf
" [ math define statement of lemma multiplyEquations(Leq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar <= metavar var u end metavar infer metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var u end metavar end define end math ] "

" [ math define proof of lemma multiplyEquations(Leq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var z end metavar infer metavar var x end metavar <= metavar var y end metavar infer metavar var z end metavar <= metavar var u end metavar infer lemma leqMultiplication modus ponens 0 <= metavar var z end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar cut lemma leqTransitivity modus ponens 0 <= metavar var x end metavar modus ponens metavar var x end metavar <= metavar var y end metavar conclude 0 <= metavar var y end metavar cut lemma leqMultiplicationLeft modus ponens 0 <= metavar var y end metavar modus ponens metavar var z end metavar <= metavar var u end metavar conclude metavar var y end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var u end metavar cut lemma leqTransitivity modus ponens metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var u end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma lessMultiplication(F) helper2 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed not0 0 <= metavar var u end metavar imply not0 not0 0 = metavar var u end metavar infer not0 0 <= metavar var v end metavar imply not0 not0 0 = metavar var v end metavar infer metavar var x end metavar <= metavar var y end metavar + - metavar var u end metavar infer 0 <= metavar var z end metavar + - metavar var v end metavar infer metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar + - metavar var u end metavar * metavar var v end metavar end define end math ] "

" [ math define proof of lemma lessMultiplication(F) helper2 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed not0 0 <= metavar var u end metavar imply not0 not0 0 = metavar var u end metavar infer not0 0 <= metavar var v end metavar imply not0 not0 0 = metavar var v end metavar infer metavar var x end metavar <= metavar var y end metavar + - metavar var u end metavar infer 0 <= metavar var z end metavar + - metavar var v end metavar infer lemma negativeToLeft(Leq) modus ponens metavar var x end metavar <= metavar var y end metavar + - metavar var u end metavar conclude metavar var x end metavar + metavar var u end metavar <= metavar var y end metavar cut axiom plusCommutativity conclude metavar var x end metavar + metavar var u end metavar = metavar var u end metavar + metavar var x end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var u end metavar = metavar var u end metavar + metavar var x end metavar modus ponens metavar var x end metavar + metavar var u end metavar <= metavar var y end metavar conclude metavar var u end metavar + metavar var x end metavar <= metavar var y end metavar cut lemma positiveToRight(Leq) modus ponens metavar var u end metavar + metavar var x end metavar <= metavar var y end metavar conclude metavar var u end metavar <= metavar var y end metavar + - metavar var x end metavar cut lemma negativeToLeft(Leq)(1 term) modus ponens 0 <= metavar var z end metavar + - metavar var v end metavar conclude metavar var v end metavar <= metavar var z end metavar cut lemma lessLeq modus ponens not0 0 <= metavar var u end metavar imply not0 not0 0 = metavar var u end metavar conclude 0 <= metavar var u end metavar cut lemma lessLeq modus ponens not0 0 <= metavar var v end metavar imply not0 not0 0 = metavar var v end metavar conclude 0 <= metavar var v end metavar cut lemma multiplyEquations(Leq) modus ponens 0 <= metavar var u end metavar modus ponens 0 <= metavar var v end metavar modus ponens metavar var u end metavar <= metavar var y end metavar + - metavar var x end metavar modus ponens metavar var v end metavar <= metavar var z end metavar conclude metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar + - metavar var x end metavar cut lemma distributionLeft conclude metavar var z end metavar * metavar var y end metavar + - metavar var x end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar cut lemma -x*y=-(x*y) conclude - metavar var x end metavar * metavar var z end metavar = - metavar var x end metavar * metavar var z end metavar cut lemma eqAdditionLeft modus ponens - metavar var x end metavar * metavar var z end metavar = - metavar var x end metavar * metavar var z end metavar conclude metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar cut lemma eqTransitivity4 modus ponens metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar + - metavar var x end metavar modus ponens metavar var z end metavar * metavar var y end metavar + - metavar var x end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar conclude metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar modus ponens metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar conclude metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar cut lemma negativeToLeft(Leq) modus ponens metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar conclude metavar var u end metavar * metavar var v end metavar + metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar cut axiom plusCommutativity conclude metavar var u end metavar * metavar var v end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar + metavar var u end metavar * metavar var v end metavar cut lemma subLeqLeft modus ponens metavar var u end metavar * metavar var v end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar + metavar var u end metavar * metavar var v end metavar modus ponens metavar var u end metavar * metavar var v end metavar + metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar conclude metavar var x end metavar * metavar var z end metavar + metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar * metavar var z end metavar cut lemma positiveToRight(Leq) modus ponens metavar var x end metavar * metavar var z end metavar + metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar * metavar var z end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar + - metavar var u end metavar * metavar var v end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessMultiplication(F) helper as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar end define end math ] "

" [ math define proof of lemma lessMultiplication(F) helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar infer for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar infer lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar cut lemma positiveFactors modus ponens not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar modus ponens not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar conclude not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar infer for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar infer metavar var n end metavar <= metavar var m end metavar infer lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar conclude metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut 1rule mp modus ponens metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar modus ponens metavar var n end metavar <= metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut 1rule mp modus ponens metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar modus ponens metavar var n end metavar <= metavar var m end metavar conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut axiom natType conclude metavar var m end metavar in0 N cut lemma 0f modus ponens metavar var m end metavar in0 N conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 cut lemma subLeqLeft modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude 0 <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut lemma lessMultiplication(F) helper2 modus ponens not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar modus ponens not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar modus ponens 0 <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut lemma timesF(Sym) conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] cut lemma subLeqLeft modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut lemma timesF(Sym) conclude [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] cut lemma eqAddition modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] conclude [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut lemma subLeqRight modus ponens [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar infer for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar infer not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar infer for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar infer metavar var n end metavar <= metavar var m end metavar infer [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut prop lemma doubly conditioned join conjuncts modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma lessMultiplication(F) as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma lessMultiplication(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep1 end metavar indeed all metavar var ep2 end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut 1rule deduction modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects metavar var ep1 end metavar indeed not0 not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut 1rule deduction modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects metavar var ep2 end metavar indeed not0 not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar cut lemma lessMultiplication(F) helper conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut pred lemma 2exist mp2 modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar imply for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar imply not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar modus ponens not0 for all objects metavar var ep1 end metavar indeed not0 not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar imply not0 not0 0 = metavar var ep1 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep1 end metavar modus ponens not0 for all objects metavar var ep2 end metavar indeed not0 not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep2 end metavar imply not0 not0 0 = metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] <= [ metavar var fz end metavar ; metavar var m end metavar ] + - metavar var ep2 end metavar conclude not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut 1rule gen modus ponens not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut pred lemma intro exist at metavar var n end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar cut pred lemma intro exist at metavar var ep1 end metavar * metavar var ep2 end metavar modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 not0 0 = metavar var ep1 end metavar * metavar var ep2 end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep1 end metavar * metavar var ep2 end metavar conclude not0 for all objects metavar var ep end metavar indeed not0 not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep end metavar cut 1rule deduction modus ponens not0 for all objects metavar var ep end metavar indeed not0 not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] + - metavar var ep end metavar conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma lessMultiplication(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma lessMultiplication(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut lemma lessMultiplication(F) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma eqReflexivity conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLessLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma eqReflexivity conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma subLessRight(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "

---------------

" [ math define statement of lemma eqMultiplicationLeft(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma eqMultiplicationLeft(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma eqMultiplication(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma timesCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma timesCommutativity(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Transitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Transitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fz end metavar ; metavar var m end metavar ] * [ metavar var fy end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma leqMultiplication(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma leqMultiplication(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqMultiplicationLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x*0=0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma x*0=0(R) conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Symmetry modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqMultiplicationLeft(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Transitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Transitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Transitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqLeq(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma eqMultiplication(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqLeq(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer prop lemma first conjunct modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma second conjunct modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut lemma lessMultiplication(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var cut lemma lessLeq(R) modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed all metavar var fz end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule repetition modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule repetition modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma expand disjuncts modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma from three disjuncts modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fz end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] <= [ metavar var fz end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] <= [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fy end metavar ; metavar var m end metavar ] * [ metavar var fz end metavar ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma 2cauchy helper as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar end define end math ] "

" [ math define proof of lemma 2cauchy helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar infer if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar infer lemma leqMax1 conclude metavar var n1 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) cut lemma leqTransitivity modus ponens metavar var n1 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar conclude metavar var n1 end metavar <= metavar var v1 end metavar cut lemma leqTransitivity modus ponens metavar var n1 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar conclude metavar var n1 end metavar <= metavar var v2 end metavar cut lemma leqMax2 conclude metavar var n2 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) cut lemma leqTransitivity modus ponens metavar var n2 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar conclude metavar var n2 end metavar <= metavar var v1 end metavar cut lemma leqTransitivity modus ponens metavar var n2 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar conclude metavar var n2 end metavar <= metavar var v2 end metavar cut lemma a4 at metavar var v1 end metavar modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma a4 at metavar var v2 end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut prop lemma mp3 modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar modus ponens metavar var n1 end metavar <= metavar var v1 end metavar modus ponens metavar var n1 end metavar <= metavar var v2 end metavar conclude not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma a4 at metavar var v1 end metavar modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma a4 at metavar var v2 end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut prop lemma mp3 modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar modus ponens metavar var n2 end metavar <= metavar var v1 end metavar modus ponens metavar var n2 end metavar <= metavar var v2 end metavar conclude not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut prop lemma join conjuncts modus ponens not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar modus ponens not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar infer if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar infer not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma 2cauchy as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar end define end math ] "

" [ math define proof of lemma 2cauchy as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed axiom cauchy conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma a4 at metavar var ep end metavar modus ponens for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut axiom cauchy conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma a4 at metavar var ep end metavar modus ponens for all objects metavar var ep end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n2 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma 2cauchy helper conclude for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut pred lemma exist mp2 modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar modus ponens not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar modus ponens not0 for all objects metavar var n2 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var v1 end metavar imply metavar var n2 end metavar <= metavar var v2 end metavar imply not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut 1rule gen modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut 1rule gen modus ponens for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut pred lemma intro exist at if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v1 end metavar imply if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut 1rule gen modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var v1 end metavar imply metavar var n end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar end quote state proof state cache var c end expand end define end math ] "











" [ math define statement of lemma addEquations(LeqLess) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar infer not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end define end math ] "

" [ math define proof of lemma addEquations(LeqLess) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar <= metavar var y end metavar infer not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar infer lemma leqAddition modus ponens metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar cut lemma lessAdditionLeft modus ponens not0 metavar var z end metavar <= metavar var u end metavar imply not0 not0 metavar var z end metavar = metavar var u end metavar conclude not0 metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar cut lemma leqLessTransitivity modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var y end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var y end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar conclude not0 metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar + metavar var u end metavar imply not0 not0 metavar var x end metavar + metavar var z end metavar = metavar var y end metavar + metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma insertTwoMiddleTerms(Numerical) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | end define end math ] "

" [ math define proof of lemma insertTwoMiddleTerms(Numerical) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed lemma insertMiddleTerm(Numerical) conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + metavar var y end metavar | cut lemma insertMiddleTerm(Numerical) conclude | metavar var z end metavar + metavar var y end metavar | <= | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | cut lemma leqAdditionLeft modus ponens | metavar var z end metavar + metavar var y end metavar | <= | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | conclude | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | cut lemma leqTransitivity modus ponens | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + metavar var y end metavar | modus ponens | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | cut axiom plusAssociativity conclude | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | conclude | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | cut lemma subLeqRight modus ponens | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | modus ponens | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var u end metavar + metavar var y end metavar | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromPositiveNumerical as system Q infer all metavar var x end metavar indeed not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | infer not0 metavar var x end metavar = 0 end define end math ] "

" [ math define proof of lemma fromPositiveNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude | metavar var x end metavar | = metavar var x end metavar cut lemma subLessRight modus ponens | metavar var x end metavar | = metavar var x end metavar modus ponens not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | conclude not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar cut lemma lessNeq modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar conclude not0 0 = metavar var x end metavar cut lemma neqSymmetry modus ponens not0 0 = metavar var x end metavar conclude not0 metavar var x end metavar = 0 cut all metavar var x end metavar indeed metavar var x end metavar <= 0 infer not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | infer lemma nonpositiveNumerical modus ponens metavar var x end metavar <= 0 conclude | metavar var x end metavar | = - metavar var x end metavar cut lemma subLessRight modus ponens | metavar var x end metavar | = - metavar var x end metavar modus ponens not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar cut lemma positiveNegated modus ponens not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar conclude not0 - - metavar var x end metavar <= 0 imply not0 not0 - - metavar var x end metavar = 0 cut lemma doubleMinus conclude - - metavar var x end metavar = metavar var x end metavar cut lemma subLessLeft modus ponens - - metavar var x end metavar = metavar var x end metavar modus ponens not0 - - metavar var x end metavar <= 0 imply not0 not0 - - metavar var x end metavar = 0 conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 cut lemma lessNeq modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 metavar var x end metavar = 0 cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed 0 <= metavar var x end metavar infer not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | infer not0 metavar var x end metavar = 0 conclude 0 <= metavar var x end metavar imply not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | imply not0 metavar var x end metavar = 0 cut 1rule deduction modus ponens all metavar var x end metavar indeed metavar var x end metavar <= 0 infer not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | infer not0 metavar var x end metavar = 0 conclude metavar var x end metavar <= 0 imply not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | imply not0 metavar var x end metavar = 0 cut not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | infer lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | imply not0 metavar var x end metavar = 0 modus ponens metavar var x end metavar <= 0 imply not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | imply not0 metavar var x end metavar = 0 conclude not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | imply not0 metavar var x end metavar = 0 cut 1rule mp modus ponens not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | imply not0 metavar var x end metavar = 0 modus ponens not0 0 <= | metavar var x end metavar | imply not0 not0 0 = | metavar var x end metavar | conclude not0 metavar var x end metavar = 0 end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma negativeToRight(Neq)(1 term) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar + - metavar var y end metavar = 0 infer not0 metavar var x end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma negativeToRight(Neq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer lemma positiveToLeft(Eq)(1 term) modus ponens metavar var x end metavar = metavar var y end metavar conclude metavar var x end metavar + - metavar var y end metavar = 0 cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer metavar var x end metavar + - metavar var y end metavar = 0 conclude metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + - metavar var y end metavar = 0 cut not0 metavar var x end metavar + - metavar var y end metavar = 0 infer prop lemma mt modus ponens metavar var x end metavar = metavar var y end metavar imply metavar var x end metavar + - metavar var y end metavar = 0 modus ponens not0 metavar var x end metavar + - metavar var y end metavar = 0 conclude not0 metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma nonzeroProduct(2) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar * metavar var y end metavar = 0 infer not0 metavar var y end metavar = 0 end define end math ] "

" [ math define proof of lemma nonzeroProduct(2) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar = 0 infer lemma eqMultiplicationLeft modus ponens metavar var y end metavar = 0 conclude metavar var x end metavar * metavar var y end metavar = metavar var x end metavar * 0 cut lemma x*0=0 conclude metavar var x end metavar * 0 = 0 cut lemma eqTransitivity modus ponens metavar var x end metavar * metavar var y end metavar = metavar var x end metavar * 0 modus ponens metavar var x end metavar * 0 = 0 conclude metavar var x end metavar * metavar var y end metavar = 0 cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var y end metavar = 0 infer metavar var x end metavar * metavar var y end metavar = 0 conclude metavar var y end metavar = 0 imply metavar var x end metavar * metavar var y end metavar = 0 cut not0 metavar var x end metavar * metavar var y end metavar = 0 infer prop lemma mt modus ponens metavar var y end metavar = 0 imply metavar var x end metavar * metavar var y end metavar = 0 modus ponens not0 metavar var x end metavar * metavar var y end metavar = 0 conclude not0 metavar var y end metavar = 0 end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma nonreciprocalToRight(Eq)(1 term) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar * metavar var y end metavar = 1 infer metavar var x end metavar = 1/ metavar var y end metavar end define end math ] "


" [ math define proof of lemma nonreciprocalToRight(Eq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar * metavar var y end metavar = 1 infer lemma eqMultiplication modus ponens metavar var x end metavar * metavar var y end metavar = 1 conclude metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = 1 * 1/ metavar var y end metavar cut lemma 0<1 conclude not0 0 <= 1 imply not0 not0 0 = 1 cut lemma positiveNonzero modus ponens not0 0 <= 1 imply not0 not0 0 = 1 conclude not0 1 = 0 cut lemma eqSymmetry modus ponens metavar var x end metavar * metavar var y end metavar = 1 conclude 1 = metavar var x end metavar * metavar var y end metavar cut lemma subNeqLeft modus ponens 1 = metavar var x end metavar * metavar var y end metavar modus ponens not0 1 = 0 conclude not0 metavar var x end metavar * metavar var y end metavar = 0 cut lemma nonzeroProduct(2) modus ponens not0 metavar var x end metavar * metavar var y end metavar = 0 conclude not0 metavar var y end metavar = 0 cut lemma x=x*y*(1/y) modus ponens not0 metavar var y end metavar = 0 conclude metavar var x end metavar = metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar cut lemma times1Left conclude 1 * 1/ metavar var y end metavar = 1/ metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar = metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar modus ponens metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = 1 * 1/ metavar var y end metavar modus ponens 1 * 1/ metavar var y end metavar = 1/ metavar var y end metavar conclude metavar var x end metavar = 1/ metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma nonreciprocalToRight(Eq) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var y end metavar = 0 infer metavar var x end metavar * metavar var y end metavar = metavar var z end metavar infer metavar var x end metavar = metavar var z end metavar * 1/ metavar var y end metavar end define end math ] "


" [ math define proof of lemma nonreciprocalToRight(Eq) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var y end metavar = 0 infer metavar var x end metavar * metavar var y end metavar = metavar var z end metavar infer lemma eqMultiplication modus ponens metavar var x end metavar * metavar var y end metavar = metavar var z end metavar conclude metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var z end metavar * 1/ metavar var y end metavar cut lemma x=x*y*(1/y) modus ponens not0 metavar var y end metavar = 0 conclude metavar var x end metavar = metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar cut lemma eqTransitivity modus ponens metavar var x end metavar = metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar modus ponens metavar var x end metavar * metavar var y end metavar * 1/ metavar var y end metavar = metavar var z end metavar * 1/ metavar var y end metavar conclude metavar var x end metavar = metavar var z end metavar * 1/ metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma nonreciprocalToLeft(Eq)(1 term) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 1 = metavar var x end metavar * metavar var y end metavar infer 1/ metavar var y end metavar = metavar var x end metavar end define end math ] "

" [ math define proof of lemma nonreciprocalToLeft(Eq)(1 term) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 1 = metavar var x end metavar * metavar var y end metavar infer lemma eqSymmetry modus ponens 1 = metavar var x end metavar * metavar var y end metavar conclude metavar var x end metavar * metavar var y end metavar = 1 cut lemma nonreciprocalToRight(Eq)(1 term) modus ponens metavar var x end metavar * metavar var y end metavar = 1 conclude metavar var x end metavar = 1/ metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = 1/ metavar var y end metavar conclude 1/ metavar var y end metavar = metavar var x end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma sameReciprocal as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = 0 infer metavar var x end metavar = metavar var y end metavar infer 1/ metavar var x end metavar = 1/ metavar var y end metavar end define end math ] "

" [ math define proof of lemma sameReciprocal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar = 0 infer metavar var x end metavar = metavar var y end metavar infer lemma times1Left conclude 1 * metavar var x end metavar = metavar var x end metavar cut lemma eqTransitivity modus ponens 1 * metavar var x end metavar = metavar var x end metavar modus ponens metavar var x end metavar = metavar var y end metavar conclude 1 * metavar var x end metavar = metavar var y end metavar cut lemma nonreciprocalToRight(Eq) modus ponens not0 metavar var x end metavar = 0 modus ponens 1 * metavar var x end metavar = metavar var y end metavar conclude 1 = metavar var y end metavar * 1/ metavar var x end metavar cut axiom timesCommutativity conclude metavar var y end metavar * 1/ metavar var x end metavar = 1/ metavar var x end metavar * metavar var y end metavar cut lemma eqTransitivity modus ponens 1 = metavar var y end metavar * 1/ metavar var x end metavar modus ponens metavar var y end metavar * 1/ metavar var x end metavar = 1/ metavar var x end metavar * metavar var y end metavar conclude 1 = 1/ metavar var x end metavar * metavar var y end metavar cut lemma nonreciprocalToLeft(Eq)(1 term) modus ponens 1 = 1/ metavar var x end metavar * metavar var y end metavar conclude 1/ metavar var y end metavar = 1/ metavar var x end metavar cut lemma eqSymmetry modus ponens 1/ metavar var y end metavar = 1/ metavar var x end metavar conclude 1/ metavar var x end metavar = 1/ metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "



" [ math define statement of lemma orderedPairEquality as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed all metavar var sz1 end metavar indeed all metavar var su end metavar indeed all metavar var su1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair comma zermelo pair metavar var sy end metavar comma metavar var sy1 end metavar end pair end pair infer zermelo pair zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair comma zermelo pair metavar var sz end metavar comma metavar var sz1 end metavar end pair end pair = zermelo pair zermelo pair metavar var su end metavar comma metavar var su end metavar end pair comma zermelo pair metavar var su end metavar comma metavar var su1 end metavar end pair end pair infer metavar var sx end metavar = metavar var sz end metavar infer metavar var sy end metavar = metavar var su end metavar end define end math ] "

" [ math define proof of lemma orderedPairEquality as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed all metavar var sz1 end metavar indeed all metavar var su end metavar indeed all metavar var su1 end metavar indeed zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair comma zermelo pair metavar var sy end metavar comma metavar var sy1 end metavar end pair end pair infer zermelo pair zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair comma zermelo pair metavar var sz end metavar comma metavar var sz1 end metavar end pair end pair = zermelo pair zermelo pair metavar var su end metavar comma metavar var su end metavar end pair comma zermelo pair metavar var su end metavar comma metavar var su1 end metavar end pair end pair infer metavar var sx end metavar = metavar var sz end metavar infer lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sx1 end metavar end pair end pair = zermelo pair zermelo pair metavar var sy end metavar comma metavar var sy end metavar end pair comma zermelo pair metavar var sy end metavar comma metavar var sy1 end metavar end pair end pair conclude metavar var sx end metavar = metavar var sy end metavar cut lemma eqSymmetry modus ponens metavar var sx end metavar = metavar var sy end metavar conclude metavar var sy end metavar = metavar var sx end metavar cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair metavar var sz end metavar comma metavar var sz end metavar end pair comma zermelo pair metavar var sz end metavar comma metavar var sz1 end metavar end pair end pair = zermelo pair zermelo pair metavar var su end metavar comma metavar var su end metavar end pair comma zermelo pair metavar var su end metavar comma metavar var su1 end metavar end pair end pair conclude metavar var sz end metavar = metavar var su end metavar cut lemma eqTransitivity4 modus ponens metavar var sy end metavar = metavar var sx end metavar modus ponens metavar var sx end metavar = metavar var sz end metavar modus ponens metavar var sz end metavar = metavar var su end metavar conclude metavar var sy end metavar = metavar var su end metavar end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma reciprocalIsFunction as system Q infer all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var end define end math ] "

" [ math define proof of lemma reciprocalIsFunction as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set infer object var var f1 end var = object var var f3 end var infer 1rule repetition modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut lemma separation2formula(2) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut 1rule repetition modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut lemma separation2formula(2) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer prop lemma first conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude object var var f1 end var = metavar var m end metavar cut lemma eqSymmetry modus ponens object var var f1 end var = metavar var m end metavar conclude metavar var m end metavar = object var var f1 end var cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude object var var f3 end var = metavar var m end metavar cut lemma eqTransitivity4 modus ponens metavar var m end metavar = object var var f1 end var modus ponens object var var f1 end var = object var var f3 end var modus ponens object var var f3 end var = metavar var m end metavar conclude metavar var m end metavar = metavar var m end metavar cut lemma sameSeries modus ponens metavar var m end metavar = metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma sameReciprocal modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude object var var f2 end var = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude object var var f4 end var = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens object var var f4 end var = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] conclude 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = object var var f4 end var cut lemma eqTransitivity4 modus ponens object var var f2 end var = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = object var var f4 end var conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer prop lemma first conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair cut prop lemma first conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut lemma orderedPairEquality modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair modus ponens object var var f1 end var = object var var f3 end var conclude metavar var m end metavar = metavar var m end metavar cut lemma sameSeries modus ponens metavar var m end metavar = metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma from contradiction modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = 0 modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer prop lemma first conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut prop lemma first conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair conclude zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair cut lemma orderedPairEquality modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair modus ponens object var var f1 end var = object var var f3 end var conclude metavar var m end metavar = metavar var m end metavar cut lemma sameSeries modus ponens metavar var m end metavar = metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqSymmetry modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut prop lemma from contradiction modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = 0 modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer prop lemma second conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut prop lemma second conjunct modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude object var var f2 end var = 0 cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude object var var f4 end var = 0 cut lemma eqSymmetry modus ponens object var var f4 end var = 0 conclude 0 = object var var f4 end var cut lemma eqTransitivity modus ponens object var var f2 end var = 0 modus ponens 0 = object var var f4 end var conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer object var var f2 end var = object var var f4 end var conclude object var var f1 end var = object var var f3 end var imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var f2 end var = object var var f4 end var cut 1rule mp modus ponens object var var f1 end var = object var var f3 end var imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens object var var f1 end var = object var var f3 end var conclude not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var f2 end var = object var var f4 end var cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer object var var f2 end var = object var var f4 end var conclude object var var f1 end var = object var var f3 end var imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var cut 1rule mp modus ponens object var var f1 end var = object var var f3 end var imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens object var var f1 end var = object var var f3 end var conclude not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair infer object var var f2 end var = object var var f4 end var conclude object var var f1 end var = object var var f3 end var imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var f2 end var = object var var f4 end var cut 1rule mp modus ponens object var var f1 end var = object var var f3 end var imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens object var var f1 end var = object var var f3 end var conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var f2 end var = object var var f4 end var cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer object var var f2 end var = object var var f4 end var conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var cut prop lemma from two times two disjuncts modus ponens not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair modus ponens not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var conclude object var var f2 end var = object var var f4 end var cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed object var var f1 end var = object var var f3 end var infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair infer object var var f2 end var = object var var f4 end var conclude object var var f1 end var = object var var f3 end var imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var cut 1rule mp modus ponens object var var f1 end var = object var var f3 end var imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens object var var f1 end var = object var var f3 end var conclude not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var cut pred lemma exist mp2 modus ponens not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair imply object var var f2 end var = object var var f4 end var modus ponens not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair modus ponens not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude object var var f2 end var = object var var f4 end var cut all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m end metavar indeed all metavar var fx end metavar indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set infer zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set infer object var var f1 end var = object var var f3 end var infer object var var f2 end var = object var var f4 end var conclude for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma reciprocalIsTotal as system Q infer all metavar var fx end metavar indeed for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set end define end math ] "

" [ math define proof of lemma reciprocalIsTotal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed [ metavar var fx end metavar ; object var var s1 end var ] = 0 infer object var var s1 end var in0 N infer axiom rationalType conclude 0 in0 Q cut lemma toCartProd modus ponens object var var s1 end var in0 N modus ponens 0 in0 Q conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma eqReflexivity conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair cut prop lemma join conjuncts modus ponens [ metavar var fx end metavar ; object var var s1 end var ] = 0 modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair conclude not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair cut prop lemma weaken or first modus ponens not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair conclude not0 not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair imply not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair cut pred lemma intro exist at object var var s1 end var modus ponens not0 not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair imply not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair conclude not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut lemma formula2separation modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut pred lemma intro exist at 0 modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 infer object var var s1 end var in0 N infer axiom seriesType conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar cut lemma valueType modus ponens object var var s1 end var in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 metavar var fx end metavar imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 metavar var fx end metavar imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 metavar var fx end metavar imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 metavar var fx end metavar conclude [ metavar var fx end metavar ; object var var s1 end var ] in0 Q cut lemma QisClosed(reciprocal) modus ponens not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 modus ponens [ metavar var fx end metavar ; object var var s1 end var ] in0 Q conclude 1/ [ metavar var fx end metavar ; object var var s1 end var ] in0 Q cut lemma toCartProd modus ponens object var var s1 end var in0 N modus ponens 1/ [ metavar var fx end metavar ; object var var s1 end var ] in0 Q conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set cut lemma eqReflexivity conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair cut prop lemma join conjuncts modus ponens not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair conclude not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair cut prop lemma weaken or second modus ponens not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair conclude not0 not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair imply not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair cut pred lemma intro exist at object var var s1 end var modus ponens not0 not0 not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair imply not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 0 end pair end pair conclude not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair cut lemma formula2separation modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set modus ponens not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair conclude zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut pred lemma intro exist at 1/ [ metavar var fx end metavar ; object var var s1 end var ] modus ponens zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma 1/ [ metavar var fx end metavar ; object var var s1 end var ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed [ metavar var fx end metavar ; object var var s1 end var ] = 0 infer object var var s1 end var in0 N infer not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut 1rule deduction modus ponens all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 infer object var var s1 end var in0 N infer not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut prop lemma from negations modus ponens [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set modus ponens not0 [ metavar var fx end metavar ; object var var s1 end var ] = 0 imply for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set end quote state proof state cache var c end expand end define end math ] "






" [ math define statement of lemma reciprocalIsRationalSeries as system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set end define end math ] "

" [ math define proof of lemma reciprocalIsRationalSeries as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma CPseparationIsRelation conclude for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair cut lemma reciprocalIsFunction conclude for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var cut lemma reciprocalIsTotal conclude for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut lemma toSeries modus ponens for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair modus ponens for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma reciprocalF nonzero as system Q infer all metavar var m end metavar indeed all metavar var m1 end metavar indeed all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 infer [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end define end math ] "


" [ math define proof of lemma reciprocalF nonzero as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var m1 end metavar indeed all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 infer axiom natType conclude metavar var m end metavar in0 N cut lemma reciprocalIsRationalSeries conclude not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut lemma memberOfSeries modus ponens metavar var m end metavar in0 N modus ponens not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut 1rule repetition modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set cut lemma separation2formula(2) modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair in0 the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set conclude not0 for all objects metavar var m1 end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair cut all metavar var m end metavar indeed all metavar var m1 end metavar indeed all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair infer prop lemma to negated and(1) modus ponens not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 conclude not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair cut prop lemma negate second disjunct modus ponens not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair conclude not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair cut prop lemma second conjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair conclude zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair cut lemma fromOrderedPair(2) modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] cut lemma fromOrderedPair(1) modus ponens zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair conclude metavar var m end metavar = metavar var m1 end metavar cut lemma sameSeries modus ponens metavar var m end metavar = metavar var m1 end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m1 end metavar ] cut lemma eqSymmetry modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m1 end metavar ] conclude [ metavar var fx end metavar ; metavar var m1 end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma sameReciprocal modus ponens not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 modus ponens [ metavar var fx end metavar ; metavar var m1 end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] conclude 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] modus ponens 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var m1 end metavar indeed all metavar var fx end metavar indeed not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 infer not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair infer [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut 1rule mp modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut pred lemma exist mp modus ponens not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens not0 for all objects metavar var m1 end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 1/ [ metavar var fx end metavar ; metavar var m1 end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m1 end metavar ] = 0 imply not0 zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair = zermelo pair zermelo pair metavar var m1 end metavar comma metavar var m1 end metavar end pair comma zermelo pair metavar var m1 end metavar comma 0 end pair end pair conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma eventually=f to sameF helper as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar end define end math ] "

" [ math define proof of lemma eventually=f to sameF helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer metavar var n end metavar <= metavar var m end metavar infer 1rule mp modus ponens metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens metavar var n end metavar <= metavar var m end metavar conclude [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma positiveToLeft(Eq)(1 term) modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = 0 cut lemma sameNumerical modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = 0 conclude | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = | 0 | cut lemma |0|=0 conclude | 0 | = 0 cut lemma eqTransitivity modus ponens | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = | 0 | modus ponens | 0 | = 0 conclude | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 0 cut lemma eqSymmetry modus ponens | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 0 conclude 0 = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut lemma subLessLeft modus ponens 0 = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar conclude not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar infer metavar var n end metavar <= metavar var m end metavar infer not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma eventually=f to sameF as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end define end math ] "

" [ math define proof of lemma eventually=f to sameF as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] infer lemma eventually=f to sameF helper conclude for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut pred lemma exist mp modus ponens for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut 1rule gen modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut pred lemma intro exist at metavar var n end metavar modus ponens for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut 1rule gen modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar cut 1rule deduction modus ponens for all objects metavar var ep end metavar indeed not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var m end metavar imply not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = metavar var ep end metavar conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule repetition modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma positiveToRight(Less) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer not0 metavar var x end metavar <= metavar var z end metavar + - metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar + - metavar var y end metavar end define end math ] "

" [ math define proof of lemma positiveToRight(Less) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed not0 metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var y end metavar = metavar var z end metavar infer lemma lessAddition modus ponens not0 metavar var x end metavar + metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + metavar var y end metavar = metavar var z end metavar conclude not0 metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar <= metavar var z end metavar + - metavar var y end metavar imply not0 not0 metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var z end metavar + - metavar var y end metavar cut lemma x=x+y-y conclude metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar = metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar cut lemma subLessLeft modus ponens metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var x end metavar modus ponens not0 metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar <= metavar var z end metavar + - metavar var y end metavar imply not0 not0 metavar var x end metavar + metavar var y end metavar + - metavar var y end metavar = metavar var z end metavar + - metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar + - metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar + - metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma switchTerms(x-y
" [ math define proof of lemma switchTerms(x-y
" [ math define statement of lemma switchTerms(x
" [ math define proof of lemma switchTerms(x

" [ math define statement of lemma insertTwoMiddleTerms(Sum) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar end define end math ] "

" [ math define proof of lemma insertTwoMiddleTerms(Sum) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed lemma insertMiddleTerm(Sum) conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar cut lemma insertMiddleTerm(Sum) conclude metavar var z end metavar + metavar var y end metavar = metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar cut lemma eqAdditionLeft modus ponens metavar var z end metavar + metavar var y end metavar = metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar conclude metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar cut axiom plusAssociativity conclude metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar conclude metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar cut lemma eqTransitivity4 modus ponens metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar modus ponens metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar modus ponens metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + - metavar var u end metavar + metavar var u end metavar + metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromNumericalGreater as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | infer not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end define end math ] "

" [ math define proof of lemma fromNumericalGreater as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | infer 0 <= metavar var y end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var y end metavar conclude | metavar var y end metavar | = metavar var y end metavar cut lemma subLessRight modus ponens | metavar var y end metavar | = metavar var y end metavar modus ponens not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | conclude not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut prop lemma weaken or first modus ponens not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | infer metavar var y end metavar <= 0 infer lemma nonpositiveNumerical modus ponens metavar var y end metavar <= 0 conclude | metavar var y end metavar | = - metavar var y end metavar cut lemma subLessRight modus ponens | metavar var y end metavar | = - metavar var y end metavar modus ponens not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | conclude not0 metavar var x end metavar <= - metavar var y end metavar imply not0 not0 metavar var x end metavar = - metavar var y end metavar cut lemma lessNegated modus ponens not0 metavar var x end metavar <= - metavar var y end metavar imply not0 not0 metavar var x end metavar = - metavar var y end metavar conclude not0 - - metavar var y end metavar <= - metavar var x end metavar imply not0 not0 - - metavar var y end metavar = - metavar var x end metavar cut lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma subLessLeft modus ponens - - metavar var y end metavar = metavar var y end metavar modus ponens not0 - - metavar var y end metavar <= - metavar var x end metavar imply not0 not0 - - metavar var y end metavar = - metavar var x end metavar conclude not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar cut prop lemma weaken or second modus ponens not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar conclude not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | infer 0 <= metavar var y end metavar infer not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | imply 0 <= metavar var y end metavar imply not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | infer metavar var y end metavar <= 0 infer not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | imply metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | infer 1rule mp modus ponens not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | imply 0 <= metavar var y end metavar imply not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | conclude 0 <= metavar var y end metavar imply not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut 1rule mp modus ponens not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | imply metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar <= | metavar var y end metavar | imply not0 not0 metavar var x end metavar = | metavar var y end metavar | conclude metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar cut lemma from leqGeq modus ponens 0 <= metavar var y end metavar imply not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar modus ponens metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar conclude not0 not0 metavar var y end metavar <= - metavar var x end metavar imply not0 not0 metavar var y end metavar = - metavar var x end metavar imply not0 metavar var x end metavar <= metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromNot



" [ math define proof of lemma fromNot
" [ math define statement of lemma fromNot
" [ math define proof of lemma fromNot



" [ math define statement of lemma fromNot
" [ math define proof of lemma fromNot
" [ math define statement of lemma fromNot
" [ math define proof of lemma fromNot
" [ math define statement of lemma fromNot
" [ math define proof of lemma fromNot
" [ math define statement of lemma fromNotSameF(Strongest) helper2 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= 1/ 1 + 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = 1/ 1 + 1 + 1 * metavar var v end metavar infer not0 | metavar var z end metavar + - metavar var u end metavar | <= 1/ 1 + 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var z end metavar + - metavar var u end metavar | = 1/ 1 + 1 + 1 * metavar var v end metavar infer metavar var v end metavar <= | metavar var y end metavar + - metavar var u end metavar | infer not0 1/ 1 + 1 + 1 * metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | imply not0 not0 1/ 1 + 1 + 1 * metavar var v end metavar = | metavar var x end metavar + - metavar var z end metavar | end define end math ] "

" [ math define proof of lemma fromNotSameF(Strongest) helper2 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed not0 | metavar var x end metavar + - metavar var y end metavar | <= 1/ 1 + 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = 1/ 1 + 1 + 1 * metavar var v end metavar infer not0 | metavar var z end metavar + - metavar var u end metavar | <= 1/ 1 + 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var z end metavar + - metavar var u end metavar | = 1/ 1 + 1 + 1 * metavar var v end metavar infer metavar var v end metavar <= | metavar var y end metavar + - metavar var u end metavar | infer lemma numericalDifference conclude | metavar var x end metavar + - metavar var y end metavar | = | metavar var y end metavar + - metavar var x end metavar | cut lemma subLessLeft modus ponens | metavar var x end metavar + - metavar var y end metavar | = | metavar var y end metavar + - metavar var x end metavar | modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= 1/ 1 + 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = 1/ 1 + 1 + 1 * metavar var v end metavar conclude not0 | metavar var y end metavar + - metavar var x end metavar | <= 1/ 1 + 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var y end metavar + - metavar var x end metavar | = 1/ 1 + 1 + 1 * metavar var v end metavar cut lemma lessNegated modus ponens not0 | metavar var y end metavar + - metavar var x end metavar | <= 1/ 1 + 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var y end metavar + - metavar var x end metavar | = 1/ 1 + 1 + 1 * metavar var v end metavar conclude not0 - 1/ 1 + 1 + 1 * metavar var v end metavar <= - | metavar var y end metavar + - metavar var x end metavar | imply not0 not0 - 1/ 1 + 1 + 1 * metavar var v end metavar = - | metavar var y end metavar + - metavar var x end metavar | cut lemma lessNegated modus ponens not0 | metavar var z end metavar + - metavar var u end metavar | <= 1/ 1 + 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var z end metavar + - metavar var u end metavar | = 1/ 1 + 1 + 1 * metavar var v end metavar conclude not0 - 1/ 1 + 1 + 1 * metavar var v end metavar <= - | metavar var z end metavar + - metavar var u end metavar | imply not0 not0 - 1/ 1 + 1 + 1 * metavar var v end metavar = - | metavar var z end metavar + - metavar var u end metavar | cut lemma addEquations(LeqLess) modus ponens metavar var v end metavar <= | metavar var y end metavar + - metavar var u end metavar | modus ponens not0 - 1/ 1 + 1 + 1 * metavar var v end metavar <= - | metavar var y end metavar + - metavar var x end metavar | imply not0 not0 - 1/ 1 + 1 + 1 * metavar var v end metavar = - | metavar var y end metavar + - metavar var x end metavar | conclude not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar <= | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | imply not0 not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar = | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | cut lemma addEquations(Less) modus ponens not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar <= | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | imply not0 not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar = | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | modus ponens not0 - 1/ 1 + 1 + 1 * metavar var v end metavar <= - | metavar var z end metavar + - metavar var u end metavar | imply not0 not0 - 1/ 1 + 1 + 1 * metavar var v end metavar = - | metavar var z end metavar + - metavar var u end metavar | conclude not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar <= | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | + - | metavar var z end metavar + - metavar var u end metavar | imply not0 not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar = | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | + - | metavar var z end metavar + - metavar var u end metavar | cut lemma insertTwoMiddleTerms(Numerical) conclude | metavar var y end metavar + - metavar var u end metavar | <= | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | cut axiom plusAssociativity conclude | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | = | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | cut axiom plusCommutativity conclude | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | = | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var y end metavar + - metavar var x end metavar | cut lemma eqTransitivity modus ponens | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | = | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | modus ponens | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | = | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var y end metavar + - metavar var x end metavar | conclude | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | = | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var y end metavar + - metavar var x end metavar | cut lemma subLeqRight modus ponens | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | = | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var y end metavar + - metavar var x end metavar | modus ponens | metavar var y end metavar + - metavar var u end metavar | <= | metavar var y end metavar + - metavar var x end metavar | + | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | conclude | metavar var y end metavar + - metavar var u end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var y end metavar + - metavar var x end metavar | cut lemma positiveToLeft(Leq) modus ponens | metavar var y end metavar + - metavar var u end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | + | metavar var y end metavar + - metavar var x end metavar | conclude | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | cut lemma positiveToLeft(Leq) modus ponens | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | <= | metavar var x end metavar + - metavar var z end metavar | + | metavar var z end metavar + - metavar var u end metavar | conclude | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | + - | metavar var z end metavar + - metavar var u end metavar | <= | metavar var x end metavar + - metavar var z end metavar | cut lemma lessLeqTransitivity modus ponens not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar <= | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | + - | metavar var z end metavar + - metavar var u end metavar | imply not0 not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar = | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | + - | metavar var z end metavar + - metavar var u end metavar | modus ponens | metavar var y end metavar + - metavar var u end metavar | + - | metavar var y end metavar + - metavar var x end metavar | + - | metavar var z end metavar + - metavar var u end metavar | <= | metavar var x end metavar + - metavar var z end metavar | conclude not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | imply not0 not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar = | metavar var x end metavar + - metavar var z end metavar | cut lemma (1/3)x+(1/3)x+(1/3)x=x conclude 1/ 1 + 1 + 1 * metavar var v end metavar + 1/ 1 + 1 + 1 * metavar var v end metavar + 1/ 1 + 1 + 1 * metavar var v end metavar = metavar var v end metavar cut lemma positiveToRight(Eq) modus ponens 1/ 1 + 1 + 1 * metavar var v end metavar + 1/ 1 + 1 + 1 * metavar var v end metavar + 1/ 1 + 1 + 1 * metavar var v end metavar = metavar var v end metavar conclude 1/ 1 + 1 + 1 * metavar var v end metavar + 1/ 1 + 1 + 1 * metavar var v end metavar = metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar cut lemma positiveToRight(Eq) modus ponens 1/ 1 + 1 + 1 * metavar var v end metavar + 1/ 1 + 1 + 1 * metavar var v end metavar = metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar conclude 1/ 1 + 1 + 1 * metavar var v end metavar = metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar cut lemma eqSymmetry modus ponens 1/ 1 + 1 + 1 * metavar var v end metavar = metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar conclude metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar = 1/ 1 + 1 + 1 * metavar var v end metavar cut lemma subLessLeft modus ponens metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar = 1/ 1 + 1 + 1 * metavar var v end metavar modus ponens not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | imply not0 not0 metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 + 1 * metavar var v end metavar = | metavar var x end metavar + - metavar var z end metavar | conclude not0 1/ 1 + 1 + 1 * metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | imply not0 not0 1/ 1 + 1 + 1 * metavar var v end metavar = | metavar var x end metavar + - metavar var z end metavar | end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma fromNotSameF(Strongest) helper as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | end define end math ] "



" [ math define proof of lemma fromNotSameF(Strongest) helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar infer metavar var n2 end metavar <= metavar var m end metavar infer prop lemma from negated double imply modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut prop lemma first conjunct modus ponens not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar conclude metavar var n1 end metavar <= metavar var n2 end metavar cut prop lemma second conjunct modus ponens not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut lemma fromNotLess modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar conclude for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var m end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar cut lemma a4 at metavar var n2 end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var m end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar conclude not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var m end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar cut lemma 0<1/3 conclude not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 cut lemma positiveFactors modus ponens not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar conclude not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar cut lemma leqTransitivity modus ponens metavar var n1 end metavar <= metavar var n2 end metavar modus ponens metavar var n2 end metavar <= metavar var m end metavar conclude metavar var n1 end metavar <= metavar var m end metavar cut prop lemma mp3 modus ponens not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var m end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar modus ponens not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar modus ponens metavar var n1 end metavar <= metavar var m end metavar modus ponens metavar var n1 end metavar <= metavar var n2 end metavar conclude not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar cut prop lemma first conjunct modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar conclude not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar cut prop lemma second conjunct modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar conclude not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar cut lemma fromNotSameF(Strongest) helper2 modus ponens not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar modus ponens not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar modus ponens metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | conclude not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar infer prop lemma from negated imply modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 not0 metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 not0 metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar cut lemma 0<1/3 conclude not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 cut lemma positiveFactors modus ponens not0 0 <= 1/ 1 + 1 + 1 imply not0 not0 0 = 1/ 1 + 1 + 1 modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar conclude not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar infer metavar var n2 end metavar <= metavar var m end metavar infer not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar infer not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar cut prop lemma doubly conditioned join conjuncts modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | end quote state proof state cache var c end expand end define end math ] "






" [ math define statement of lemma fromNotSameF(Strongest) as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 for all objects metavar var ep end metavar indeed not0 not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | end define end math ] "

" [ math define proof of lemma fromNotSameF(Strongest) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule deduction modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut pred lemma AEAnegated modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut lemma 2cauchy conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma a4 at 1/ 1 + 1 + 1 * metavar var ep end metavar modus ponens for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar cut lemma fromNotSameF(Strongest) helper conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut pred lemma EAE mp modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut pred lemma exist mp modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 + 1 * metavar var ep end metavar conclude not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut 1rule gen modus ponens not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude for all objects metavar var m end metavar indeed not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut pred lemma intro exist at metavar var n2 end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n2 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut pred lemma intro exist at 1/ 1 + 1 + 1 * metavar var ep end metavar modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 1/ 1 + 1 + 1 * metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 1/ 1 + 1 + 1 * metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 for all objects metavar var ep end metavar indeed not0 not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma toLess(F) helper as system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar imply not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var m end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar end define end math ] "

" [ math define proof of lemma toLess(F) helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar infer if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var m end metavar infer lemma leqMax1 conclude metavar var n1 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) cut lemma leqTransitivity modus ponens metavar var n1 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var m end metavar conclude metavar var n1 end metavar <= metavar var m end metavar cut lemma leqMax2 conclude metavar var n2 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) cut lemma leqTransitivity modus ponens metavar var n2 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var m end metavar conclude metavar var n2 end metavar <= metavar var m end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut 1rule mp modus ponens metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens metavar var n1 end metavar <= metavar var m end metavar conclude not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut lemma fromNumericalGreater modus ponens not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] <= - metavar var ep end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = - metavar var ep end metavar imply not0 metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut prop lemma mp2 modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar modus ponens metavar var n2 end metavar <= metavar var m end metavar conclude not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar cut lemma lessNegated modus ponens not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar conclude not0 - metavar var ep end metavar <= - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] imply not0 not0 - metavar var ep end metavar = - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma minusNegated conclude - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma subLessRight modus ponens - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens not0 - metavar var ep end metavar <= - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] imply not0 not0 - metavar var ep end metavar = - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] conclude not0 - metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 - metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma lessLeq modus ponens not0 - metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 - metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] conclude - metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma toNotLess modus ponens - metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] conclude not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] <= - metavar var ep end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = - metavar var ep end metavar cut prop lemma negate first disjunct modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] <= - metavar var ep end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = - metavar var ep end metavar imply not0 metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] <= - metavar var ep end metavar imply not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] = - metavar var ep end metavar conclude not0 metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma switchTerms(x



" [ math define statement of lemma toLess(F) as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma toLess(F) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer 1rule repetition modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut prop lemma from negated or modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut prop lemma first conjunct modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma second conjunct modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut lemma fromNot

" [ math define statement of lemma from!!== as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end define end math ] "

" [ math define proof of lemma from!!== as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer 1rule to== modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer prop lemma mt modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma toLess(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var end define end math ] "

" [ math define proof of lemma toLess(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer 1rule repetition modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma from negated or modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut prop lemma first conjunct modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut prop lemma second conjunct modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma fromNot<< modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut lemma from!!== modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut prop lemma join conjuncts modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut prop lemma to negated or modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply not0 not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule repetition modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut lemma toLess(F) modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var imply for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var cut 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma fromNotLess(R) as system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma fromNotLess(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma toLess(R) modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var cut not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var infer prop lemma negative mt modus ponens not0 not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set imply not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var modus ponens not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fy end metavar ; object var var m end var ] + - object var var ep end var conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fy end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fy end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma fromNotSameF(Strong) helper2 as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | infer not0 | metavar var x end metavar + - metavar var y end metavar | <= 1/ 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = 1/ 1 + 1 * metavar var v end metavar infer not0 | metavar var z end metavar + - metavar var u end metavar | <= 1/ 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var z end metavar + - metavar var u end metavar | = 1/ 1 + 1 * metavar var v end metavar infer not0 metavar var y end metavar = metavar var u end metavar end define end math ] "

" [ math define proof of lemma fromNotSameF(Strong) helper2 as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed all metavar var u end metavar indeed all metavar var v end metavar indeed metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | infer not0 | metavar var x end metavar + - metavar var y end metavar | <= 1/ 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = 1/ 1 + 1 * metavar var v end metavar infer not0 | metavar var z end metavar + - metavar var u end metavar | <= 1/ 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var z end metavar + - metavar var u end metavar | = 1/ 1 + 1 * metavar var v end metavar infer lemma lessNegated modus ponens not0 | metavar var x end metavar + - metavar var y end metavar | <= 1/ 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var x end metavar + - metavar var y end metavar | = 1/ 1 + 1 * metavar var v end metavar conclude not0 - 1/ 1 + 1 * metavar var v end metavar <= - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 - 1/ 1 + 1 * metavar var v end metavar = - | metavar var x end metavar + - metavar var y end metavar | cut lemma numericalDifference conclude | metavar var z end metavar + - metavar var u end metavar | = | metavar var u end metavar + - metavar var z end metavar | cut lemma subLessLeft modus ponens | metavar var z end metavar + - metavar var u end metavar | = | metavar var u end metavar + - metavar var z end metavar | modus ponens not0 | metavar var z end metavar + - metavar var u end metavar | <= 1/ 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var z end metavar + - metavar var u end metavar | = 1/ 1 + 1 * metavar var v end metavar conclude not0 | metavar var u end metavar + - metavar var z end metavar | <= 1/ 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var u end metavar + - metavar var z end metavar | = 1/ 1 + 1 * metavar var v end metavar cut lemma lessNegated modus ponens not0 | metavar var u end metavar + - metavar var z end metavar | <= 1/ 1 + 1 * metavar var v end metavar imply not0 not0 | metavar var u end metavar + - metavar var z end metavar | = 1/ 1 + 1 * metavar var v end metavar conclude not0 - 1/ 1 + 1 * metavar var v end metavar <= - | metavar var u end metavar + - metavar var z end metavar | imply not0 not0 - 1/ 1 + 1 * metavar var v end metavar = - | metavar var u end metavar + - metavar var z end metavar | cut lemma addEquations(Less) modus ponens not0 - 1/ 1 + 1 * metavar var v end metavar <= - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 - 1/ 1 + 1 * metavar var v end metavar = - | metavar var x end metavar + - metavar var y end metavar | modus ponens not0 - 1/ 1 + 1 * metavar var v end metavar <= - | metavar var u end metavar + - metavar var z end metavar | imply not0 not0 - 1/ 1 + 1 * metavar var v end metavar = - | metavar var u end metavar + - metavar var z end metavar | conclude not0 - 1/ 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 * metavar var v end metavar <= - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | imply not0 not0 - 1/ 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 * metavar var v end metavar = - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | cut lemma (1/2)x+(1/2)x=x conclude 1/ 1 + 1 * metavar var v end metavar + 1/ 1 + 1 * metavar var v end metavar = metavar var v end metavar cut lemma eqNegated modus ponens 1/ 1 + 1 * metavar var v end metavar + 1/ 1 + 1 * metavar var v end metavar = metavar var v end metavar conclude - 1/ 1 + 1 * metavar var v end metavar + 1/ 1 + 1 * metavar var v end metavar = - metavar var v end metavar cut lemma -x-y=-(x+y) conclude - 1/ 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 * metavar var v end metavar = - 1/ 1 + 1 * metavar var v end metavar + 1/ 1 + 1 * metavar var v end metavar cut lemma eqTransitivity modus ponens - 1/ 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 * metavar var v end metavar = - 1/ 1 + 1 * metavar var v end metavar + 1/ 1 + 1 * metavar var v end metavar modus ponens - 1/ 1 + 1 * metavar var v end metavar + 1/ 1 + 1 * metavar var v end metavar = - metavar var v end metavar conclude - 1/ 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 * metavar var v end metavar = - metavar var v end metavar cut lemma subLessLeft modus ponens - 1/ 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 * metavar var v end metavar = - metavar var v end metavar modus ponens not0 - 1/ 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 * metavar var v end metavar <= - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | imply not0 not0 - 1/ 1 + 1 * metavar var v end metavar + - 1/ 1 + 1 * metavar var v end metavar = - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | conclude not0 - metavar var v end metavar <= - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | imply not0 not0 - metavar var v end metavar = - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | cut axiom plusCommutativity conclude - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | = - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | cut lemma subLessRight modus ponens - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | = - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | modus ponens not0 - metavar var v end metavar <= - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | imply not0 not0 - metavar var v end metavar = - | metavar var x end metavar + - metavar var y end metavar | + - | metavar var u end metavar + - metavar var z end metavar | conclude not0 - metavar var v end metavar <= - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 - metavar var v end metavar = - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | cut lemma addEquations(LeqLess) modus ponens metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | modus ponens not0 - metavar var v end metavar <= - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 - metavar var v end metavar = - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | conclude not0 metavar var v end metavar + - metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 metavar var v end metavar + - metavar var v end metavar = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | cut axiom negative conclude metavar var v end metavar + - metavar var v end metavar = 0 cut lemma subLessLeft modus ponens metavar var v end metavar + - metavar var v end metavar = 0 modus ponens not0 metavar var v end metavar + - metavar var v end metavar <= | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 metavar var v end metavar + - metavar var v end metavar = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | conclude not0 0 <= | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 0 = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | cut axiom plusAssociativity conclude | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | conclude | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | cut lemma subLessRight modus ponens | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | modus ponens not0 0 <= | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 0 = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | conclude not0 0 <= | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 0 = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | cut lemma insertTwoMiddleTerms(Numerical) conclude | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + - metavar var y end metavar | + | metavar var y end metavar + - metavar var u end metavar | + | metavar var u end metavar + - metavar var z end metavar | cut lemma positiveToLeft(Leq) modus ponens | metavar var x end metavar + - metavar var z end metavar | <= | metavar var x end metavar + - metavar var y end metavar | + | metavar var y end metavar + - metavar var u end metavar | + | metavar var u end metavar + - metavar var z end metavar | conclude | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | <= | metavar var x end metavar + - metavar var y end metavar | + | metavar var y end metavar + - metavar var u end metavar | cut axiom plusCommutativity conclude | metavar var x end metavar + - metavar var y end metavar | + | metavar var y end metavar + - metavar var u end metavar | = | metavar var y end metavar + - metavar var u end metavar | + | metavar var x end metavar + - metavar var y end metavar | cut lemma subLeqRight modus ponens | metavar var x end metavar + - metavar var y end metavar | + | metavar var y end metavar + - metavar var u end metavar | = | metavar var y end metavar + - metavar var u end metavar | + | metavar var x end metavar + - metavar var y end metavar | modus ponens | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | <= | metavar var x end metavar + - metavar var y end metavar | + | metavar var y end metavar + - metavar var u end metavar | conclude | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | <= | metavar var y end metavar + - metavar var u end metavar | + | metavar var x end metavar + - metavar var y end metavar | cut lemma positiveToLeft(Leq) modus ponens | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | <= | metavar var y end metavar + - metavar var u end metavar | + | metavar var x end metavar + - metavar var y end metavar | conclude | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | <= | metavar var y end metavar + - metavar var u end metavar | cut lemma lessLeqTransitivity modus ponens not0 0 <= | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | imply not0 not0 0 = | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | modus ponens | metavar var x end metavar + - metavar var z end metavar | + - | metavar var u end metavar + - metavar var z end metavar | + - | metavar var x end metavar + - metavar var y end metavar | <= | metavar var y end metavar + - metavar var u end metavar | conclude not0 0 <= | metavar var y end metavar + - metavar var u end metavar | imply not0 not0 0 = | metavar var y end metavar + - metavar var u end metavar | cut lemma fromPositiveNumerical modus ponens not0 0 <= | metavar var y end metavar + - metavar var u end metavar | imply not0 not0 0 = | metavar var y end metavar + - metavar var u end metavar | conclude not0 metavar var y end metavar + - metavar var u end metavar = 0 cut lemma negativeToRight(Neq)(1 term) modus ponens not0 metavar var y end metavar + - metavar var u end metavar = 0 conclude not0 metavar var y end metavar = metavar var u end metavar end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma fromNotSameF(Strong) helper as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] end define end math ] "


" [ math define proof of lemma fromNotSameF(Strong) helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar infer metavar var n2 end metavar <= metavar var m end metavar infer prop lemma from negated double imply modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut prop lemma first conjunct modus ponens not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar conclude metavar var n1 end metavar <= metavar var n2 end metavar cut prop lemma second conjunct modus ponens not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var n2 end metavar imply not0 not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut lemma fromNotLess modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | cut lemma a4 at metavar var n2 end metavar modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar conclude for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar cut lemma a4 at metavar var m end metavar modus ponens for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar conclude not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply metavar var n1 end metavar <= metavar var m end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar cut lemma positiveHalved modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar conclude not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar cut lemma leqTransitivity modus ponens metavar var n1 end metavar <= metavar var n2 end metavar modus ponens metavar var n2 end metavar <= metavar var m end metavar conclude metavar var n1 end metavar <= metavar var m end metavar cut prop lemma mp3 modus ponens not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply metavar var n1 end metavar <= metavar var m end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar modus ponens not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar modus ponens metavar var n1 end metavar <= metavar var n2 end metavar modus ponens metavar var n1 end metavar <= metavar var m end metavar conclude not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar cut prop lemma first conjunct modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar conclude not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar cut prop lemma second conjunct modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar conclude not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar cut lemma fromNotSameF(Strong) helper2 modus ponens metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | modus ponens not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar modus ponens not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar infer for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar infer metavar var n2 end metavar <= metavar var m end metavar infer not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "




" [ math define statement of lemma fromNotSameF(Strong) as system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] end define end math ] "


" [ math define proof of lemma fromNotSameF(Strong) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var v1 end metavar indeed all metavar var v2 end metavar indeed all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var infer 1rule repetition modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var cut 1rule deduction modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ metavar var fy end metavar ; object var var m end var ] | = object var var ep end var conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut pred lemma AEAnegated modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar cut lemma 2cauchy conclude for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar cut lemma a4 at 1/ 1 + 1 * metavar var ep end metavar modus ponens for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = metavar var ep end metavar conclude not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar cut lemma fromNotSameF(Strong) helper conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut pred lemma EAE mp modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar imply for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens not0 for all objects metavar var ep end metavar indeed not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var n2 end metavar indeed not0 not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n1 end metavar <= metavar var n2 end metavar imply not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n2 end metavar ] + - [ metavar var fy end metavar ; metavar var n2 end metavar ] | = metavar var ep end metavar conclude for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut pred lemma exist mp modus ponens for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens not0 for all objects metavar var n1 end metavar indeed not0 for all objects metavar var v1 end metavar indeed for all objects metavar var v2 end metavar indeed not0 0 <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 0 = 1/ 1 + 1 * metavar var ep end metavar imply metavar var n1 end metavar <= metavar var v1 end metavar imply metavar var n1 end metavar <= metavar var v2 end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var v1 end metavar ] + - [ metavar var fx end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | <= 1/ 1 + 1 * metavar var ep end metavar imply not0 not0 | [ metavar var fy end metavar ; metavar var v1 end metavar ] + - [ metavar var fy end metavar ; metavar var v2 end metavar ] | = 1/ 1 + 1 * metavar var ep end metavar conclude metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut 1rule gen modus ponens metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude for all objects metavar var m end metavar indeed metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] cut pred lemma intro exist at metavar var n2 end metavar modus ponens for all objects metavar var m end metavar indeed metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fy end metavar ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "





" [ math define statement of lemma sameFreciprocal helper as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] imply metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end define end math ] "

" [ math define proof of lemma sameFreciprocal helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] infer metavar var n end metavar <= metavar var m end metavar infer lemma a4 at metavar var m end metavar modus ponens for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] conclude metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] cut 1rule mp modus ponens metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] modus ponens metavar var n end metavar <= metavar var m end metavar conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] cut axiom natType conclude metavar var m end metavar in0 N cut lemma 0f modus ponens metavar var m end metavar in0 N conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 cut lemma subNeqRight modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] = 0 modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] conclude not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 cut lemma reciprocalF nonzero modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma eqMultiplicationLeft modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = 1/ [ metavar var fx end metavar ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * 1/ [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma reciprocal modus ponens not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 conclude [ metavar var fx end metavar ; metavar var m end metavar ] * 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = 1 cut lemma 1f modus ponens metavar var m end metavar in0 N conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] = 1 cut lemma eqSymmetry modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] = 1 conclude 1 = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] cut lemma eqTransitivity modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = 1 modus ponens 1 = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] conclude [ metavar var fx end metavar ; metavar var m end metavar ] * 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] cut lemma timesF conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] cut lemma eqTransitivity4 modus ponens [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] * 1/ [ metavar var fx end metavar ; metavar var m end metavar ] modus ponens [ metavar var fx end metavar ; metavar var m end metavar ] * 1/ [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] conclude [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var fx end metavar indeed for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] infer metavar var n end metavar <= metavar var m end metavar infer [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] conclude for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] imply metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] end quote state proof state cache var c end expand end define end math ] "


" [ math define statement of lemma sameFreciprocal as system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | = object var var ep end var infer for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] | = object var var ep end var end define end math ] "

" [ math define proof of lemma sameFreciprocal as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | = object var var ep end var infer lemma fromNotSameF(Strong) modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | = object var var ep end var conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] cut lemma sameFreciprocal helper conclude for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] imply metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] cut pred lemma exist mp modus ponens for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] imply metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; metavar var m end metavar ] conclude metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] cut 1rule gen modus ponens metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] conclude for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] cut pred lemma intro exist at metavar var n end metavar modus ponens metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] conclude not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] cut lemma eventually=f to sameF modus ponens not0 for all objects metavar var n end metavar indeed not0 for all objects metavar var m end metavar indeed metavar var n end metavar <= metavar var m end metavar imply [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; metavar var m end metavar ] = [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; metavar var m end metavar ] conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] | = object var var ep end var end quote state proof state cache var c end expand end define end math ] "

-------------




" [ math define statement of lemma leqReflexivity(R) as system Q infer all metavar var fx end metavar indeed not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma leqReflexivity(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed lemma eqReflexivity conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqLeq(R) modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 not0 for all objects object var var ep end var indeed not0 not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply not0 object var var n end var <= object var var m end var imply [ metavar var fx end metavar ; object var var m end var ] <= [ metavar var fx end metavar ; object var var m end var ] + - object var var ep end var imply the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "

" [ math define statement of lemma reciprocal(R) as system Q infer all metavar var fx end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end define end math ] "

" [ math define proof of lemma reciprocal(R) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var fx end metavar indeed not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set infer lemma from!!== modus ponens not0 the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | = object var var ep end var cut lemma sameFreciprocal modus ponens not0 for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ metavar var fx end metavar ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 0 end pair end pair end set ; object var var m end var ] | = object var var ep end var conclude for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] | = object var var ep end var cut 1rule to== modus ponens for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] | = object var var ep end var conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma eqReflexivity conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut lemma ==Transitivity modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set cut 1rule repetition modus ponens the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set conclude the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 placeholder-var var e end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma [ metavar var fx end metavar ; metavar var m end metavar ] * [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects metavar var m end metavar indeed not0 not0 not0 not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 1/ [ metavar var fx end metavar ; metavar var m end metavar ] end pair end pair imply not0 [ metavar var fx end metavar ; metavar var m end metavar ] = 0 imply not0 placeholder-var var f end var = zermelo pair zermelo pair metavar var m end metavar comma metavar var m end metavar end pair comma zermelo pair metavar var m end metavar comma 0 end pair end pair end set ; metavar var m end metavar ] end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set = the set of ph in power the set of ph in power the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set end power such that not0 not0 for all objects object var var r1 end var indeed object var var r1 end var in0 placeholder-var var f end var imply not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 object var var r1 end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair imply not0 for all objects object var var f1 end var indeed for all objects object var var f2 end var indeed for all objects object var var f3 end var indeed for all objects object var var f4 end var indeed zermelo pair zermelo pair object var var f1 end var comma object var var f1 end var end pair comma zermelo pair object var var f1 end var comma object var var f2 end var end pair end pair in0 placeholder-var var f end var imply zermelo pair zermelo pair object var var f3 end var comma object var var f3 end var end pair comma zermelo pair object var var f3 end var comma object var var f4 end var end pair end pair in0 placeholder-var var f end var imply object var var f1 end var = object var var f3 end var imply object var var f2 end var = object var var f4 end var imply not0 for all objects object var var s1 end var indeed object var var s1 end var in0 N imply not0 for all objects object var var s2 end var indeed not0 zermelo pair zermelo pair object var var s1 end var comma object var var s1 end var end pair comma zermelo pair object var var s1 end var comma object var var s2 end var end pair end pair in0 placeholder-var var f end var end set end power such that for all objects object var var ep end var indeed not0 for all objects object var var n end var indeed not0 for all objects object var var m end var indeed not0 0 <= object var var ep end var imply not0 not0 0 = object var var ep end var imply object var var n end var <= object var var m end var imply not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | <= object var var ep end var imply not0 not0 | [ the set of ph in the set of ph in power power U( zermelo pair N comma Q end pair ) end power end power such that not0 for all objects object var var op1 end var indeed not0 not0 for all objects object var var op2 end var indeed not0 not0 not0 object var var op1 end var in0 N imply not0 object var var op2 end var in0 Q imply not0 placeholder-var var a end var = zermelo pair zermelo pair object var var op1 end var comma object var var op1 end var end pair comma zermelo pair object var var op1 end var comma object var var op2 end var end pair end pair end set such that not0 for all objects object var var crs1 end var indeed not0 placeholder-var var c end var = zermelo pair zermelo pair object var var crs1 end var comma object var var crs1 end var end pair comma zermelo pair object var var crs1 end var comma 1 end pair end pair end set ; object var var m end var ] + - [ placeholder-var var d end var ; object var var m end var ] | = object var var ep end var end set end quote state proof state cache var c end expand end define end math ] "



venter----


\appendix

" [ flush left math define priority of sup as 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 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 \end{flushleft}

\newpage

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item " [ math define tex of sup as text "sup" end text end define end math ] "

\item " [ math define tex of var x ^ var y as text "(#1.
(expARGH!) #2.
)" end text end define end math ] "




\item " [ math define tex of lemma nonreciprocalToRight(Eq)(1 term) as text "NonreciprocalToRight(Eq)(1 term)" end text end define end math ] "

\item " [ math define tex of lemma plusAssociativity(4 terms) as text "PlusAssociativity(4 terms)" end text end define end math ] "


\item " [ math define tex of lemma nonzeroProduct(2) as text "NonzeroProduct(2)" end text end define end math ] "

\item " [ math define tex of lemma eqLeq(R) as text "eqLeq(R)" end text end define end math ] "

\item " [ math define tex of lemma thirdGeqSeries as text "ThirdGeqSeries" end text end define end math ] "

\item " [ math define tex of lemma negativeToLeft(Leq) as text "negativeToLeft(Leq)" end text end define end math ] "

\item " [ math define tex of lemma negativeToLeft(Leq)(1 term) as text "negativeToLeft(Leq)(1 term)" end text end define end math ] "

\item " [ math define tex of lemma UStelescope(+1) as text "UStelescope(+1)" end text end define end math ] "

\item " [ math define tex of lemma telescopeBound base as text "TelescopeBound(Base)" end text end define end math ] "

\item " [ math define tex of lemma telescopeBound indu as text "TelescopeBound(Indu)" end text end define end math ] "

\item " [ math define tex of lemma telescopeBound as text "TelescopeBound" end text end define end math ] "

\item " [ math define tex of lemma intervalSize base as text "IntervalSize(Base)" end text end define end math ] "

\item " [ math define tex of lemma intervalSize indu as text "IntervalSize(Indu)" end text end define end math ] "

\item " [ math define tex of lemma intervalSize as text "IntervalSize" end text end define end math ] "

\item " [ math define tex of lemma XSlessUS as text "XS<US" end text end define end math ] "



\item " [ math define tex of lemma closeUS as text "CloseUS" end text end define end math ] "

\item " [ math define tex of lemma closeUS(n+1) as text "CloseUS(n+1)" end text end define end math ] "

\item " [ math define tex of lemma induction as text "Induction" end text end define end math ] "

\item " [ math define tex of lemma leqAntisymmetry as text "leqAntisymmetry" end text end define end math ] "

\item " [ math define tex of lemma leqTransitivity as text "leqTransitivity" end text end define end math ] "

\item " [ math define tex of lemma leqAddition as text "leqAddition" end text end define end math ] "

\item " [ math define tex of lemma reciprocal as text "Reciprocal" end text end define end math ] "

\item " [ math define tex of lemma equality as text "Equality" end text end define end math ] "

\item " [ math define tex of lemma eqLeq as text "eqLeq" end text end define end math ] "

\item " [ math define tex of lemma eqAddition as text "eqAddition" end text end define end math ] "


\item " [ math define tex of lemma eqMultiplication as text "eqMultiplication" end text end define end math ] "

\item " [ math define tex of lemma eqReflexivity as text "eqReflexivity" end text end define end math ] "

\item " [ math define tex of lemma eqSymmetry as text "eqSymmetry" end text end define end math ] "

\item " [ math define tex of lemma eqTransitivity as text "eqTransitivity" end text end define end math ] "

\item " [ math define tex of lemma eqTransitivity4 as text "eqTransitivity4" end text end define end math ] "

\item " [ math define tex of lemma eqTransitivity5 as text "eqTransitivity5" end text end define end math ] "

\item " [ math define tex of lemma eqTransitivity6 as text "eqTransitivity6" end text end define end math ] "

\item " [ math define tex of lemma plus0Left as text "plus0Left" end text end define end math ] "

\item " [ math define tex of lemma times1Left as text "times1Left" end text end define end math ] "

\item " [ math define tex of lemma eqMultiplicationLeft as text "EqMultiplicationLeft" end text end define end math ] "

\item " [ math define tex of lemma distributionLeft as text "DistributionLeft" end text end define end math ] "

\item " [ math define tex of lemma distributionOut as text "DistributionOut" end text end define end math ] "

\item " [ math define tex of lemma distributionOutLeft as text "DistributionOutLeft" end text end define end math ] "

\item " [ math define tex of lemma three2twoTerms as text "Three2twoTerms" end text end define end math ] "

\item " [ math define tex of lemma three2threeTerms as text "Three2threeTerms" end text end define end math ] "

\item " [ math define tex of lemma three2twoFactors as text "Three2twoFactors" end text end define end math ] "

\item " [ math define tex of lemma three2threeFactors as text "Three2threeFactors" end text end define end math ] "

\item " [ math define tex of lemma addEquations as text "AddEquations" end text end define end math ] "

\item " [ math define tex of lemma subtractEquations as text "SubtractEquations" end text end define end math ] "

\item " [ math define tex of lemma subtractEquationsLeft as text "SubtractEquationsLeft" end text end define end math ] "

\item " [ math define tex of lemma multiplyEquations as text "MultiplyEquations" end text end define end math ] "

\item " [ math define tex of lemma eqNegated as text "EqNegated" end text end define end math ] "

\item " [ math define tex of lemma positiveToRight(Eq) as text "PositiveToRight(Eq)" end text end define end math ] "

\item " [ math define tex of lemma positiveToLeft(Eq)(1 term) as text "PositiveToLeft(Eq)(1 term)" end text end define end math ] "

\item " [ math define tex of lemma negativeToLeft(Eq) as text "NegativeToLeft(Eq)" end text end define end math ] "

\item " [ math define tex of lemma reciprocalToLeft(Less) as text "reciprocalToLeft(Less)" end text end define end math ] "

\item " [ math define tex of lemma lessNeq as text "LessNeq" end text end define end math ] "

\item " [ math define tex of lemma neqSymmetry as text "NeqSymmetry" end text end define end math ] "

\item " [ math define tex of lemma neqNegated as text "NeqNegated" end text end define end math ] "

\item " [ math define tex of lemma subNeqRight as text "SubNeqRight" end text end define end math ] "

\item " [ math define tex of lemma subNeqLeft as text "SubNeqLeft" end text end define end math ] "

\item " [ math define tex of lemma negativeToRight(Neq)(1 term) as text "NegativeToRight(Neq)(1 term)" end text end define end math ] "

\item " [ math define tex of lemma neqAddition as text "NeqAddition" end text end define end math ] "

\item " [ math define tex of lemma neqMultiplication as text "NeqMultiplication" end text end define end math ] "

\item " [ math define tex of lemma uniqueNegative as text "UniqueNegative" end text end define end math ] "

\item " [ math define tex of lemma doubleMinus as text "DoubleMinus" end text end define end math ] "

\item " [ math define tex of lemma leqMultiplicationLeft as text "LeqMultiplicationLeft " end text end define end math ] "

\item " [ math define tex of lemma leqLessEq as text "LeqLessEq" end text end define end math ] "

\item " [ math define tex of lemma lessLeq as text "LessLeq" end text end define end math ] "

\item " [ math define tex of lemma from leqGeq as text "FromLeqGeq" end text end define end math ] "

\item " [ math define tex of lemma subLeqRight as text "subLeqRight" end text end define end math ] "

\item " [ math define tex of lemma subLeqLeft as text "subLeqLeft" end text end define end math ] "

\item " [ math define tex of lemma leqPlus1 as text "Leq+1" end text end define end math ] "

\item " [ math define tex of lemma positiveToRight(Leq) as text "PositiveToRight(Leq)" end text end define end math ] "

\item " [ math define tex of lemma positiveToRight(Leq)(1 term) as text "PositiveToRight(Leq)(1 term)" end text end define end math ] "

\item " [ math define tex of lemma positiveToLeft(Leq) as text "PositiveToLeft(Leq)" end text end define end math ] "

\item " [ math define tex of lemma leqAdditionLeft as text "LeqAdditionLeft" end text end define end math ] "

\item " [ math define tex of lemma leqSubtraction as text "leqSubtraction" end text end define end math ] "

\item " [ math define tex of lemma leqSubtractionLeft as text "leqSubtractionLeft" end text end define end math ] "

\item " [ math define tex of lemma leqMultiplication as text "leqMultiplication" end text end define end math ] "

\item " [ math define tex of lemma thirdGeq as text "thirdGeq" end text end define end math ] "

\item " [ math define tex of lemma leqNegated as text "LeqNegated" end text end define end math ] "

\item " [ math define tex of lemma addEquations(Leq) as text "AddEquations(Leq)" end text end define end math ] "

\item " [ math define tex of lemma multiplyEquations(Leq) as text "MultiplyEquations(Leq)" end text end define end math ] "

\item " [ math define tex of lemma leqNeqLess as text "LeqNeqLess" end text end define end math ] "

\item " [ math define tex of lemma fromLess as text "FromLess" end text end define end math ] "

\item " [ math define tex of lemma toLess as text "ToLess" end text end define end math ] "

\item " [ math define tex of lemma fromNotLess as text "fromNotLess" end text end define end math ] "

\item " [ math define tex of lemma toNotLess as text "toNotLess" end text end define end math ] "

\item " [ math define tex of lemma lessAddition as text "LessAddition" end text end define end math ] "

\item " [ math define tex of lemma lessAdditionLeft as text "LessAdditionLeft" end text end define end math ] "

\item " [ math define tex of lemma lessMultiplication as text "LessMultiplication" end text end define end math ] "


\item " [ math define tex of lemma lessMultiplicationLeft as text "LessMultiplicationLeft" end text end define end math ] "

\item " [ math define tex of lemma lessDivision as text "LessDivision" end text end define end math ] "

\item " [ math define tex of lemma positiveToRight(Less) as text "PositiveToRight(Less)" end text end define end math ] "

\item " [ math define tex of lemma positiveToLeft(Less) as text "PositiveToLeft(Less)" end text end define end math ] "

\item " [ math define tex of lemma negativeToLeft(Less) as text "NegativeToLeft(Less)" end text end define end math ] "

\item " [ math define tex of lemma negativeToRight(Less) as text "NegativeToRight(Less)" end text end define end math ] "

\item " [ math define tex of lemma addEquations(Less) as text "AddEquations(Less)" end text end define end math ] "

\item " [ math define tex of lemma addEquations(LeqLess) as text "AddEquations(LeqLess)" end text end define end math ] "

\item " [ math define tex of lemma negativeLessPositive as text "NegativeLessPositive" end text end define end math ] "

\item " [ math define tex of lemma leqLessTransitivity as text "leqLessTransitivity" end text end define end math ] "

\item " [ math define tex of lemma lessLeqTransitivity as text "LessLeqTransitivity" end text end define end math ] "

\item " [ math define tex of lemma lessTransitivity as text "LessTransitivity" end text end define end math ] "

\item " [ math define tex of lemma lessTotality as text "LessTotality" end text end define end math ] "

\item " [ math define tex of lemma subLessRight as text "SubLessRight" end text end define end math ] "

\item " [ math define tex of lemma subLessLeft as text "SubLessLeft" end text end define end math ] "

\item " [ math define tex of lemma switchTerms(x
\item " [ math define tex of lemma switchTerms(x-y
\item " [ math define tex of lemma lessNegated as text "LessNegated" end text end define end math ] "

\item " [ math define tex of lemma positiveNonzero as text "PositiveNonzero" end text end define end math ] "

\item " [ math define tex of lemma positiveNegated as text "PositiveNegated" end text end define end math ] "

\item " [ math define tex of lemma nonpositiveNegated as text "NonpositiveNegated" end text end define end math ] "

\item " [ math define tex of lemma negativeNegated as text "NegativeNegated" end text end define end math ] "

\item " [ math define tex of lemma nonnegativeNegated as text "NonnegativeNegated" end text end define end math ] "

\item " [ math define tex of lemma positiveInverted as text "PositiveInverted" end text end define end math ] "

\item " [ math define tex of lemma positiveHalved as text "PositiveHalved" end text end define end math ] "

\item " [ math define tex of lemma nonnegativeNumerical as text "NonnegativeNumerical" end text end define end math ] "

\item " [ math define tex of lemma negativeNumerical as text "NegativeNumerical" end text end define end math ] "

\item " [ math define tex of lemma positiveNumerical as text "PositiveNumerical" end text end define end math ] "

\item " [ math define tex of lemma |0|=0 as text "|0|=0" end text end define end math ] "

\item " [ math define tex of lemma 0<=|x| as text "0<=|x|" end text end define end math ] "

\item " [ math define tex of lemma x<=|x| as text "x<=|x|" end text end define end math ] "

\item " [ math define tex of lemma fromPositiveNumerical as text "FromPositiveNumerical" end text end define end math ] "

\item " [ math define tex of lemma sameNumerical as text "SameNumerical" end text end define end math ] "

\item " [ math define tex of lemma signNumerical(+) as text "SignNumerical(+)" end text end define end math ] "

\item " [ math define tex of lemma signNumerical as text "SignNumerical" end text end define end math ] "

\item " [ math define tex of lemma toNumericalLess as text "ToNumericalLess" end text end define end math ] "

\item " [ math define tex of lemma fromNumericalGreater as text "FromNumericalGreater" end text end define end math ] "

\item " [ math define tex of lemma numericalDifference as text "NumericalDifference" end text end define end math ] "

\item " [ math define tex of lemma numericalDifferenceLess helper as text "NumericalDifferenceLess(Helper)" end text end define end math ] "

\item " [ math define tex of lemma numericalDifferenceLess as text "NumericalDifferenceLess" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalSumHelper as text "SplitNumericalSumHelper" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalSum(++) as text "splitNumericalSum(++)" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalSum(--) as text "splitNumericalSum(--)" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalSum(+-, smallNegative) as text "splitNumericalSum(+-small)" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalSum(+-, bigNegative) as text "splitNumericalSum(+-big)" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalSum(+-) as text "splitNumericalSum(+-)" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalSum(-+) as text "splitNumericalSum(-+)" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalSum as text "splitNumericalSum" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalProduct(++) as text "SplitNumericalProduct(++)" end text end define end math ] "


\item " [ math define tex of lemma splitNumericalProduct(+-) as text "SplitNumericalProduct(+-)" end text end define end math ] "

\item " [ math define tex of lemma splitNumericalProduct as text "SplitNumericalProduct" end text end define end math ] "

\item " [ math define tex of lemma insertMiddleTerm(Numerical) as text "insertMiddleTerm(Numerical)" end text end define end math ] "

\item " [ math define tex of lemma insertTwoMiddleTerms(Numerical) as text "insertTwoMiddleTerms(Numerical)" end text end define end math ] "

\item " [ math define tex of lemma leqMax1 as text "MaxLeq(1)" end text end define end math ] "

\item " [ math define tex of lemma leqMax2 as text "MaxLeq(2)" end text end define end math ] "

\item " [ math define tex of lemma lessThanMax as text "LessThanMax" end text end define end math ] "

\item " [ math define tex of lemma x+y=zBackwards as text "x+y=zBackwards" end text end define end math ] "

\item " [ math define tex of lemma x*y=zBackwards as text "x*y=zBackwards" end text end define end math ] "

\item " [ math define tex of lemma x=x+(y-y) as text "x=x+(y-y)" end text end define end math ] "

\item " [ math define tex of lemma x=x+y-y as text "x=x+y-y" end text end define end math ] "

\item " [ math define tex of lemma x=x*y*(1/y) as text "x=x*y*(1/y)" end text end define end math ] "

\item " [ math define tex of lemma insertMiddleTerm(Sum) as text "insertMiddleTerm(Sum)" end text end define end math ] "

\item " [ math define tex of lemma insertTwoMiddleTerms(Sum) as text "insertTwoMiddleTerms(Sum)" end text end define end math ] "

\item " [ math define tex of lemma insertMiddleTerm(Difference) as text "insertMiddleTerm(Difference)" end text end define end math ] "

\item " [ math define tex of lemma x*0+x=x as text "x*0+x=x" end text end define end math ] "

\item " [ math define tex of lemma nonnegativeFactors as text "NonnegativeFactors" end text end define end math ] "

\item " [ math define tex of lemma nonzeroFactors as text "NonzeroFactors" end text end define end math ] "

\item " [ math define tex of lemma positiveFactors as text "PositiveFactors" end text end define end math ] "

\item " [ math define tex of lemma plusTimesMinus as text "PlusTimesMinus" end text end define end math ] "

\item " [ math define tex of lemma minusTimesMinus as text "MinusTimesMinus" end text end define end math ] "

\item " [ math define tex of lemma x*0=0 as text "x*0=0" end text end define end math ] "

\item " [ math define tex of lemma (-1)*(-1)+(-1)*1=0 as text "(-1)*(-1)+(-1)*1=0" end text end define end math ] "

\item " [ math define tex of lemma (-1)*(-1)=1 as text "(-1)*(-1)=1" end text end define end math ] "

\item " [ math define tex of lemma 0<1Helper as text "0<1Helper" end text end define end math ] "

\item " [ math define tex of lemma 0<1 as text "0<1" end text end define end math ] "

\item " [ math define tex of lemma 0<2 as text "0<2" end text end define end math ] "

\item " [ math define tex of lemma 0<3 as text "0<3" end text end define end math ] "

\item " [ math define tex of lemma 0<1/2 as text "0<1/2" end text end define end math ] "

\item " [ math define tex of lemma 0<1/3 as text "0<1/3" end text end define end math ] "

\item " [ math define tex of lemma x+x=2*x as text "TwoWholes" end text end define end math ] "

\item " [ math define tex of lemma x+x+x=3*x as text "ThreeWholes" end text end define end math ] "

\item " [ math define tex of lemma (1/2)x+(1/2)x=x as text "TwoHalves" end text end define end math ] "

\item " [ math define tex of lemma (1/3)x+(1/3)x+(1/3)x=x as text "ThreeThirds" end text end define end math ] "

\item " [ math define tex of lemma -x-y=-(x+y) as text "-x-y=-(x+y)" end text end define end math ] "

\item " [ math define tex of lemma -x*y=-(x*y) as text "-x*y=-(x*y)" end text end define end math ] "

\item " [ math define tex of lemma minusNegated as text "MinusNegated" end text end define end math ] "

\item " [ math define tex of lemma times(-1) as text "Times(-1)" end text end define end math ] "

\item " [ math define tex of lemma times(-1)Left as text "Times(-1)Left" end text end define end math ] "

\item " [ math define tex of lemma -0=0 as text "-0=0" end text end define end math ] "

\item " [ math define tex of pred lemma allNegated(Imply) as text "AllNegated(Imply)" end text end define end math ] "

\item " [ math define tex of pred lemma existNegated(Imply) as text "ExistNegated(Imply)" end text end define end math ] "

\item " [ math define tex of pred lemma intro exist helper as text "IntroExist(Helper)" end text end define end math ] "

\item " [ math define tex of pred lemma intro exist as text "IntroExist" end text end define end math ] "

\item " [ math define tex of pred lemma exist mp as text "ExistMP" end text end define end math ] "

\item " [ math define tex of pred lemma exist mp2 as text "ExistMP2" end text end define end math ] "

\item " [ math define tex of pred lemma 2exist mp as text "TwiceExistMP" end text end define end math ] "

\item " [ math define tex of pred lemma 2exist mp2 as text "TwiceExistMP2" end text end define end math ] "

\item " [ math define tex of pred lemma EAE mp as text "EAE-MP" end text end define end math ] "

\item " [ math define tex of pred lemma addAll as text "AddAll " end text end define end math ] "

\item " [ math define tex of pred lemma addExist helper1 as text "AddExist(Helper1)" end text end define end math ] "

\item " [ math define tex of pred lemma addExist helper2 as text "AddExist(Helper2)" end text end define end math ] "

\item " [ math define tex of pred lemma addExist as text "AddExist" end text end define end math ] "

\item " [ math define tex of pred lemma addExist(SimpleAnt) as text "AddExist(SimpleAnt)" end text end define end math ] "

\item " [ math define tex of pred lemma addExist(Simple) as text "AddExist(Simple)" end text end define end math ] "

\item " [ math define tex of pred lemma addEAE as text "AddEAE" end text end define end math ] "

\item " [ math define tex of pred lemma AEAnegated as text "AEA-negated" end text end define end math ] "

\item " [ math define tex of pred lemma EEAnegated as text "EEA-negated" end text end define end math ] "


\item " [ math define tex of prop lemma to negated and as text "ToNegatedAnd" end text end define end math ] "

\item " [ math define tex of lemma eqTransitivity4 as text "eqTransitivity4" end text end define end math ] "

\item " [ math define tex of lemma sameFsymmetry as text "SFsymmetry" end text end define end math ] "

\item " [ math define tex of lemma sameFtransitivity as text "SFtransitivity" end text end define end math ] "

\item " [ math define tex of lemma plusF(Sym) as text "PlusF(Sym)" end text end define end math ] "

\item " [ math define tex of lemma timesF(Sym) as text "TimesF(Sym)" end text end define end math ] "

\item " [ math define tex of lemma f2R(Plus) as text "f2R(Plus)" end text end define end math ] "

\item " [ math define tex of lemma f2R(Times) as text "f2R(Times)" end text end define end math ] "

\item " [ math define tex of lemma <
\item " [ math define tex of lemma <
\item " [ math define tex of lemma <<==Reflexivity as text "<<==Reflexivity" end text end define end math ] "

\item " [ math define tex of lemma <<==AntisymmetryHelper(Q) as text "<<==AntisymmetryHelper(Q)" end text end define end math ] "

\item " [ math define tex of lemma fromNotSameF(Weak)(Helper) as text "FromNotSameF(Weak)(Helper)" end text end define end math ] "

\item " [ math define tex of lemma fromNotSameF(Weak) as text "FromNotSameF(Weak)" end text end define end math ] "

\item " [ math define tex of lemma fromNotLess(F) as text "FromNotLess(F)" end text end define end math ] "

\item " [ math define tex of lemma plus0(F) as text "Plus0(F)" end text end define end math ] "
\item " [ math define tex of lemma ==Addition as text "==Addition" end text end define end math ] "

\item " [ math define tex of lemma ==AdditionLeft as text "==AdditionLeft" end text end define end math ] "
\item " [ math define tex of lemma fpart-Bounded base as text "Fpart-Bounded(Base)" end text end define end math ] "

\item " [ math define tex of lemma fpart-Bounded indu helper as text "Fpart-Bounded(InduHelper)" end text end define end math ] "

\item " [ math define tex of lemma fpart-Bounded indu as text "Fpart-Bounded(Indu)" end text end define end math ] "

\item " [ math define tex of lemma fpart-Bounded as text "Fpart-Bounded" end text end define end math ] "

\item " [ math define tex of lemma f-Bounded as text "F-Bounded" end text end define end math ] "

\item " [ math define tex of lemma f-Bounded helper as text "F-Bounded(Helper)" end text end define end math ] "

\item " [ math define tex of lemma sameFmultiplication helper as text "SameFmultiplication(Helper)" end text end define end math ] "

\item " [ math define tex of lemma sameFmultiplication as text "SameFmultiplication" end text end define end math ] "

\item " [ math define tex of lemma fromNot
\item " [ math define tex of lemma fromNot
\item " [ math define tex of lemma fromNot
\item " [ math define tex of lemma fromNot
\item " [ math define tex of lemma fromNot
\item " [ math define tex of lemma fromNotSameF(Strongest) helper2 as text "fromNotSameF(Strongest)(Helper2)" end text end define end math ] "

\item " [ math define tex of lemma fromNotSameF(Strongest) helper as text "fromNotSameF(Strongest)(Helper)" end text end define end math ] "

\item " [ math define tex of lemma fromNotSameF(Strongest) as text "fromNotSameF(Strongest)" end text end define end math ] "

\item " [ math define tex of lemma toLess(F) helper as text "ToLess(F)(Helper)" end text end define end math ] "

\item " [ math define tex of lemma toLess(F) as text "ToLess(F)" end text end define end math ] "

\item " [ math define tex of lemma lessMultiplication(F) helper2 as text "LessMultiplication(F)(Helper2)" end text end define end math ] "

\item " [ math define tex of lemma lessMultiplication(F) helper as text "LessMultiplication(F)(Helper)" end text end define end math ] "

\item " [ math define tex of lemma lessMultiplication(F) as text "LessMultiplication(F)" end text end define end math ] "

\item " [ math define tex of lemma eqMultiplication(R) as text "EqMultiplication(R)" end text end define end math ] "

\item " [ math define tex of lemma eqMultiplicationLeft(R) as text "EqMultiplicationLeft(R)" end text end define end math ] "

\item " [ math define tex of lemma plusAssociativity(F) as text "PlusAssociativity(F)" end text end define end math ] "

\item " [ math define tex of lemma fromNot<< as text "FromNot<<" end text end define end math ] "

\item " [ math define tex of lemma toLess(R) as text "ToLess(R)" end text end define end math ] "

\item " [ math define tex of lemma x*0=0(F) as text "x*0=0(F)" end text end define end math ] "

\item " [ math define tex of lemma x*0=0(R) as text "x*0=0(R)" end text end define end math ] "

\item " [ math define tex of lemma plusCommutativity(F) as text "PlusCommutativity(F)" end text end define end math ] "

\item " [ math define tex of lemma 2cauchy helper as text "Cauchy(2)(Helper)" end text end define end math ] "

\item " [ math define tex of lemma 2cauchy as text "Cauchy(2)" end text end define end math ] "

\item " [ math define tex of lemma timesAssociativity(F) as text "TimesAssociativity(F)" end text end define end math ] "

\item " [ math define tex of lemma lessMultiplication(R) as text "LessMultiplication(R)" end text end define end math ] "

\item " [ math define tex of lemma leqMultiplication(R) as text "LeqMultiplication(R)" end text end define end math ] "

\item " [ math define tex of lemma times1f as text "Times1f" end text end define end math ] "

\item " [ math define tex of lemma reciprocalF nonzero as text "ReciprocalFnonzero" end text end define end math ] "

\item " [ math define tex of lemma eventually=f to sameF helper as text "(Eventually=f)2sameF(Helper)" end text end define end math ] "

\item " [ math define tex of lemma eventually=f to sameF as text "(Eventually=f)2sameF" end text end define end math ] "

\item " [ math define tex of lemma fromNotSameF(Strong) helper2 as text "FromNotSameF(Strong)(Helper2)" end text end define end math ] "
\item " [ math define tex of lemma fromNotSameF(Strong) helper as text "FromNotSameF(Strong)(Helper)" end text end define end math ] "

\item " [ math define tex of lemma fromNotSameF(Strong) as text "FromNotSameF(Strong)" end text end define end math ] "

\item " [ math define tex of lemma sameFreciprocal helper as text "SameFreciprocal(Helper)" end text end define end math ] "

\item " [ math define tex of lemma sameFreciprocal as text "SameFreciprocal" end text end define end math ] "

\item " [ math define tex of lemma from!!== as text "From!!==" end text end define end math ] "

\item " [ math define tex of lemma reciprocal(R) as text "Reciprocal(R)" end text end define end math ] "

\item " [ math define tex of lemma timesCommutativity(F) as text "TimesCommutativity(F)" end text end define end math ] "

\item " [ math define tex of lemma distribution(F) as text "Distribution(F)" end text end define end math ] "

\item " [ math define tex of lemma fromNotLess(R) as text "FromNotLess(R)" end text end define end math ] "

\item " [ math define tex of prop lemma to negated and(1) as text "ToNegatedAnd(1)" end text end define end math ] "

\item " [ math define tex of lemma fromMax(1) as text "FromMax(1)" end text end define end math ] "

\item " [ math define tex of lemma fromMax(2) as text "FromMax(2)" end text end define end math ] "


\item " [ math define tex of lemma cartProdIsRelation as text "CartProdIsRelation" end text end define end math ] "


\item " [ math define tex of lemma fromSubset as text "FromSubset" end text end define end math ] "

\item " [ math define tex of lemma subsetIsRelation as text "SubsetIsRelation" end text end define end math ] "

\item " [ math define tex of lemma seriesSubsetCP as text "SeriesSubsetCP" end text end define end math ] "

\item " [ math define tex of lemma valueType as text "ValueType" end text end define end math ] "

\item " [ math define tex of lemma toSeries as text "ToSeries" end text end define end math ] "

\item " [ math define tex of lemma fromSeries as text "FromSeries" end text end define end math ] "

\item " [ math define tex of prop lemma remove or as text "RemoveOr" end text end define end math ] "

\item " [ math define tex of lemma fromSingleton as text "FromSingleton" end text end define end math ] "

\item " [ math define tex of lemma inPair(1) as text "InPair(1)" end text end define end math ] "

\item " [ math define tex of lemma inPair(2) as text "InPair(2)" end text end define end math ] "

\item " [ math define tex of lemma sameMember(2) as text "SameMember(2)" end text end define end math ] "

\item " [ math define tex of lemma toBinaryUnion(1) as text "ToBinaryUnion(1)" end text end define end math ] "

\item " [ math define tex of lemma toBinaryUnion(2) as text "ToBinaryUnion(2)" end text end define end math ] "

\item " [ math define tex of lemma fromOrderedPair(twoLevels) as text "FromOrderedPair(TwoLevels)" end text end define end math ] "

\item " [ math define tex of lemma toCartProd helper as text "ToCartProd(Helper)" end text end define end math ] "

\item " [ math define tex of lemma toCartProd as text "ToCartProd" end text end define end math ] "

\item " [ math define tex of lemma nonreciprocalToRight(Eq) as text "NonreciprocalToRight(Eq)" end text end define end math ] "

\item " [ math define tex of lemma nonreciprocalToLeft(Eq)(1 term) as text "NonreciprocalToLeft(Eq)(1 term)" end text end define end math ] "

\item " [ math define tex of lemma sameReciprocal as text "SameReciprocal" end text end define end math ] "


\item " [ math define tex of lemma CPseparationIsRelation as text "CPseparationIsRelation" end text end define end math ] "

\item " [ math define tex of lemma orderedPairEquality as text "OrderedPairEquality" end text end define end math ] "

\item " [ math define tex of lemma reciprocalIsFunction as text "ReciprocalIsFunction" end text end define end math ] "

\item " [ math define tex of lemma reciprocalIsTotal as text "ReciprocalIsTotal" end text end define end math ] "

\item " [ math define tex of lemma reciprocalIsRationalSeries as text "ReciprocalIsRationalSeries" end text end define end math ] "

\item " [ math define tex of lemma crsIsRelation as text "CrsIsRelation" end text end define end math ] "

\item " [ math define tex of lemma crsIsFunction as text "CrsIsFunction " end text end define end math ] "

\item " [ math define tex of lemma crsIsTotal as text "CrsIsTotal" end text end define end math ] "

\item " [ math define tex of lemma crsIsSeries as text "CrsIsSeries" end text end define end math ] "

\item " [ math define tex of lemma crsLookup as text "CrsLookup" end text end define end math ] "

\item " [ math define tex of lemma 0f as text "0f" end text end define end math ] "

\item " [ math define tex of lemma 1f as text "1f" end text end define end math ] "

\item " [ math define tex of lemma toSingleton as text "ToSingleton" end text end define end math ] "

\item " [ math define tex of lemma fromSameSingleton as text "FromSameSingleton" end text end define end math ] "


\item " [ math define tex of lemma singletonmembersEqual as text "SingletonmembersEqual" end text end define end math ] "

\item " [ math define tex of lemma unequalsNotInSingleton as text "UnequalsNotInSingleton" end text end define end math ] "

\item " [ math define tex of lemma nonsingletonmembersUnequal as text "NonsingletonmembersUnequal" end text end define end math ] "

\item " [ math define tex of lemma fromOrderedPair as text "FromOrderedPair" end text end define end math ] "

\item " [ math define tex of lemma fromOrderedPair(1) as text "FromOrderedPair(1)" end text end define end math ] "

\item " [ math define tex of lemma fromOrderedPair(2) as text "FromOrderedPair(2)" end text end define end math ] "

\item " [ math define tex of lemma fromCartProd as text "FromCartProd" end text end define end math ] "

\item " [ math define tex of lemma fromCartProd(1) as text "FromCartProd(1)" end text end define end math ] "

\item " [ math define tex of lemma fromCartProd(2) as text "FromCartProd(2)" end text end define end math ] "

\item " [ math define tex of lemma sameOrderedPair as text "sameOrderedPair" end text end define end math ] "

\item " [ math define tex of lemma inSeries helper as text "InSeriesHelper" end text end define end math ] "

\item " [ math define tex of lemma inSeries as text "InSeries" end text end define end math ] "

\item " [ math define tex of lemma to=f subset helper as text "To=f(Subset)(Helper)" end text end define end math ] "

\item " [ math define tex of lemma to=f subset as text "To=f(Subset)" end text end define end math ] "

\item " [ math define tex of lemma to=f as text "To=f" end text end define end math ] "

\item " [ math define tex of tester1 as text "Tester1" end text end define end math ] "

\item " [ math define tex of tester2 as text "Tester2" end text end define end math ] "

\item " [ math define tex of tester3 as text "Tester3" end text end define end math ] "

\item " [ math define tex of tester4 as text "Tester4" end text end define end math ] "

\item " [ math define tex of tester5 as text "Tester5" end text end define end math ] "

\item " [ math define tex of tester6 as text "Tester6" end text end define end math ] "

\item " [ math define tex of lemma productIsFunction as text "productIsFunction" end text end define end math ] "

\item " [ math define tex of lemma productIsTotal as text "productIsTotal" end text end define end math ] "

\item " [ math define tex of lemma productIsRationalSeries as text "ProductIsRationalSeries" end text end define end math ] "

\item " [ math define tex of lemma timesF as text "TimesF" end text end define end math ] "

\item " [ math define tex of lemma -x+(1/2)x=-(1/2)x as text "-x+(1/2)x=-(1/2)x" end text end define end math ] "

\item " [ math define tex of lemma closetolessIsLess as text "ClosetolessIsLess" end text end define end math ] "

\item " [ math define tex of lemma subLessLeft(F) as text "SubLessLeft(F)" end text end define end math ] "

\item " [ math define tex of lemma subLessLeft(R) as text "SubLessLeft(R)" end text end define end math ] "

\item " [ math define tex of lemma closetogreaterIsGreater as text "ClosetogreaterIsGreater" end text end define end math ] "

\item " [ math define tex of lemma subLessRight(F) as text "SubLessRight(F)" end text end define end math ] "

\item " [ math define tex of lemma subLessRight(R) as text "SubLessRight(R)" end text end define end math ] "

\item " [ math define tex of lemma positiveTripled as text "PositiveTripled" end text end define end math ] "

\item " [ math define tex of lemma positiveDividedBy3 as text "PositiveDividedBy3" end text end define end math ] "

\item " [ math define tex of lemma |x-x|=0 as text "|x-x|=0" end text end define end math ] "

\item " [ math define tex of lemma 1<2 as text "1<2" end text end define end math ] "

\item " [ math define tex of lemma 1/3<2/3 as text "1/3<2/3" end text end define end math ] "

\item " [ math define tex of lemma (1/3)x+(1/3)x=(2/3)x as text "(1/3)x+(1/3)x=(2/3)x" end text end define end math ] "

\item " [ math define tex of lemma (2/3)x+(1/3)x=x as text "(2/3)x+(1/3)x=x" end text end define end math ] "

\item " [ math define tex of lemma -x+(2/3)x=-(1/3)x as text "-x+(2/3)x=-(1/3)x" end text end define end math ] "

\item " [ math define tex of lemma preserveLessGreater as text "PreserveLessGreater" end text end define end math ] "

\item " [ math define tex of lemma -(1/3)x-(1/3)x=-(2/3)x as text "-(1/3)x-(1/3)x=-(2/3)x" end text end define end math ] "

\item " [ math define tex of lemma -x+(1/3)x=-(2/3)x as text "-x+(1/3)x=-(2/3)x" end text end define end math ] "

\item " [ math define tex of lemma plus0Left as text "plus0Left" end text end define end math ] "

\item " [ math define tex of lemma times1Left as text "times1Left" end text end define end math ] "

\item " [ math define tex of lemma eqAdditionLeft as text "EqAdditionLeft" end text end define end math ] "

\item " [ math define tex of lemma eqMultiplicationLeft as text "EqMultiplicationLeft" end text end define end math ] "

\item " [ math define tex of lemma plusF(Sym) as text "PlusF(Sym)" end text end define end math ] "

\item " [ math define tex of lemma timesF(Sym) as text "TimesF(Sym)" end text end define end math ] "

\item " [ math define tex of lemma sameSeries(Gen) as text "SameSeries(Gen)" end text end define end math ] "

\item " [ math define tex of lemma equalsSameF as text "EqualsSameF" end text end define end math ] "

\item " [ math define tex of lemma leqReflexivity(R) as text "LeqReflexivity(R)" end text end define end math ] "



\end{list}

\end{document}

End of file
latex page
latex page
dvipdfm page"