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\begin {document}

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



SUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUP

$
[ SystemQ
\mathbf{\ lemma\ } ToNegatedDoubleImply
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
\colon {\cal A}
\vdash {\cal B}
\vdash \dot{\neg}\,({\cal C}
)n
\vdash \dot{\neg}\,(({\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
)
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } ToNegatedDoubleImply
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,({\cal C}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(({\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
\Rightarrow \dot{\neg}\,({\cal C}
)n
\Rightarrow ({\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
)
\Rightarrow \dot{\neg}\,(({\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,({\cal C}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$MP3
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
)
\Rightarrow \dot{\neg}\,(({\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$
\mbox{prop lemma imply negation}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(({\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
)
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } FromNegatedAnd(Imply)
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon \dot{\neg}\,(({\cal A}
\mathrel{\dot{\wedge}} {\cal B}
)
)n
\Rightarrow {\cal A}
\Rightarrow \dot{\neg}\,({\cal B}
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } FromNegatedAnd(Imply)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(({\cal A}
\mathrel{\dot{\wedge}} {\cal B}
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromNegatedAnd
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,({\cal B}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(({\cal A}
\mathrel{\dot{\wedge}} {\cal B}
)
)n
\Rightarrow {\cal A}
\Rightarrow \dot{\neg}\,({\cal B}
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } RemoveDoubleNeg(Consequent)
\colon 
\Pi {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal A}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,({\cal B}
)n
)n
\vdash {\cal A}
\Rightarrow {\cal B}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } RemoveDoubleNeg(Consequent)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,({\cal B}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(\dot{\neg}\,({\cal B}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$RemoveDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal A}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,({\cal B}
)n
)n
)
\Rightarrow {\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,({\cal B}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } (A)to(~E~)(Imply)
\colon 
\Pi V_{1}
,\linebreak [0] {\cal A}
\colon 
\forall V_{1}
\colon {\cal A}
\Rightarrow \dot{\neg}\,(
\exists V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } (A)to(~E~)(Imply)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall V_{1}
\colon {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$A4
\mathop {\char64} V_{1}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$AddDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(\dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall V_{1}
\colon \dot{\neg}\,(\dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$AddDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(\dot{\neg}\,(
\forall V_{1}
\colon \dot{\neg}\,(\dot{\neg}\,({\cal A}
)n
)n
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\exists V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall V_{1}
\colon {\cal A}
\Rightarrow \dot{\neg}\,(
\exists V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } (E)to(~A~)(Imply)
\colon 
\Pi V_{1}
,\linebreak [0] {\cal A}
\colon 
\exists V_{1}
\colon {\cal A}
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } (E)to(~A~)(Imply)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$AutoImply
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\forall V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{1}
\colon {\cal A}
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




$
[ SystemQ
\mathbf{\ lemma\ } (E~)to(~A)(Imply)
\colon 
\Pi V_{1}
,\linebreak [0] {\cal A}
\colon 
\exists V_{1}
\colon \dot{\neg}\,({\cal A}
)n
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon {\cal A}
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } (E~)to(~A)(Imply)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{1}
\colon \dot{\neg}\,({\cal A}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$AddDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(\dot{\neg}\,(
\exists V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$(A)to(~E~)(Imply)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall V_{1}
\colon {\cal A}
\Rightarrow \dot{\neg}\,(
\exists V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\forall V_{1}
\colon {\cal A}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{1}
\colon \dot{\neg}\,({\cal A}
)n
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon {\cal A}
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



$
[ SystemQ
\mathbf{\ lemma\ } AddNegatedAll
\colon 
\Pi V_{1}
,\linebreak [0] {\cal A}
,\linebreak [0] {\cal B}
\colon {\cal B}
\Rightarrow {\cal A}
\vdash \dot{\neg}\,(
\forall V_{1}
\colon {\cal A}
)n
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon {\cal B}
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } AddNegatedAll
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] {\cal A}
,\linebreak [0] {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\Rightarrow {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$AddAll 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall V_{1}
\colon {\cal B}
\Rightarrow 
\forall V_{1}
\colon {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Contrapositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\forall V_{1}
\colon {\cal A}
)n
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon {\cal B}
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



$
[ SystemQ
\mathbf{\ lemma\ } ToNegatedAEA 
\colon 
\Pi V_{1}
,\linebreak [0] V_{2}
,\linebreak [0] V_{3}
,\linebreak [0] {\cal A}
\colon 
\exists V_{1}
\colon 
\forall V_{2}
\colon 
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
\vdash \dot{\neg}\,(
\forall V_{1}
\colon 
\exists V_{2}
\colon 
\forall V_{3}
\colon {\cal A}
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } ToNegatedAEA 
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] V_{2}
,\linebreak [0] V_{3}
,\linebreak [0] {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{1}
\colon 
\forall V_{2}
\colon 
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$(E~)to(~A)(Imply)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
\Rightarrow \dot{\neg}\,(
\forall V_{3}
\colon {\cal A}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$AddNegatedAll
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\forall V_{2}
\colon \dot{\neg}\,(
\forall V_{3}
\colon {\cal A}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall V_{2}
\colon 
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$(E)to(~A~)(Imply)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{2}
\colon 
\forall V_{3}
\colon {\cal A}
\Rightarrow \dot{\neg}\,(
\forall V_{2}
\colon \dot{\neg}\,(
\forall V_{3}
\colon {\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$ImplyTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{2}
\colon 
\forall V_{3}
\colon {\cal A}
\Rightarrow \dot{\neg}\,(
\forall V_{2}
\colon 
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$AddNegatedAll
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\forall V_{1}
\colon \dot{\neg}\,(
\forall V_{2}
\colon 
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon 
\exists V_{2}
\colon 
\forall V_{3}
\colon {\cal A}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$(E)to(~A~)(Imply)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{1}
\colon 
\forall V_{2}
\colon 
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon \dot{\neg}\,(
\forall V_{2}
\colon 
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$ImplyTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{1}
\colon 
\forall V_{2}
\colon 
\exists V_{3}
\colon \dot{\neg}\,({\cal A}
)n
\Rightarrow \dot{\neg}\,(
\forall V_{1}
\colon 
\exists V_{2}
\colon 
\forall V_{3}
\colon {\cal A}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\forall V_{1}
\colon 
\exists V_{2}
\colon 
\forall V_{3}
\colon {\cal A}
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } (A~)to(~E)
\colon 
\Pi V_{1}
,\linebreak [0] {\cal A}
\colon 
\forall V_{1}
\colon \dot{\neg}\,({\cal A}
)n
\vdash \dot{\neg}\,(
\exists V_{1}
\colon {\cal A}
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } (A~)to(~E)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall V_{1}
\colon \dot{\neg}\,({\cal A}
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$AddDoubleNeg
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(\dot{\neg}\,(
\forall V_{1}
\colon \dot{\neg}\,({\cal A}
)n
)n
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\exists V_{1}
\colon {\cal A}
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } ExistMP3
\colon 
\Pi V_{1}
,\linebreak [0] V_{2}
,\linebreak [0] V_{3}
,\linebreak [0] {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
\colon {\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
\vdash 
\exists V_{1}
\colon {\cal A}
\vdash 
\exists V_{2}
\colon {\cal B}
\vdash 
\exists V_{3}
\colon {\cal C}
\vdash {\cal D}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } ExistMP3
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$V_{1}
,\linebreak [0] V_{2}
,\linebreak [0] V_{3}
,\linebreak [0] {\cal A}
,\linebreak [0] {\cal B}
,\linebreak [0] {\cal C}
,\linebreak [0] {\cal D}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\Rightarrow {\cal B}
\Rightarrow {\cal C}
\Rightarrow {\cal D}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{1}
\colon {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{2}
\colon {\cal B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists V_{3}
\colon {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$ExistMP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
\Rightarrow {\cal D}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ExistMP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal D}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $





$
[ SystemQ
\mathbf{\ lemma\ } PositiveToLeft(Eq)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
\colon {\cal X}
 = ({\cal Y}
+{\cal Z}
)
\vdash ({\cal X}
-{\cal Z}
)
 = {\cal Y}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveToLeft(Eq)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 = ({\cal Y}
+{\cal Z}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$eqAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
-{\cal Z}
)
 = (({\cal Y}
+{\cal Z}
)
-{\cal Z}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$x=x+y-y
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 = (({\cal Y}
+{\cal Z}
)
-{\cal Z}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal Y}
+{\cal Z}
)
-{\cal Z}
)
 = {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
-{\cal Z}
)
 = {\cal Y}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




--------


$
[ SystemQ
\mathbf{\ lemma\ } ExpZero(Exact) 
\colon 
\Pi {\cal X}
\colon  {\cal X}
 (exp) 0
 = 1
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } ExpZero(Exact) 
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$eqReflexivity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ExpZero
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 0
 = 1
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } (+1)IsPositive(N)
\colon 
\Pi {\cal M}
\colon Nat({\cal M}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 0
 < ({\cal M}
+1
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } (+1)IsPositive(N)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{%
$\mbox{Side-condition}{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$Nat({\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Nonnegative(N)
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <= {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Leq+1
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ({\cal M}
+1
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $ 


$
[ SystemQ
\mathbf{\ lemma\ } SameExp(Base)
\colon 
\Pi {\cal N}
,\linebreak [0] {\cal X}
\colon 
\forall {\cal N}
\colon 
(0
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 0
 =  {\cal X}
 (exp) {\cal N}
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameExp(Base)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal N}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$ExpZero(Exact) 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 0
 = 1
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal N}
 = 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$ExpZero
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) {\cal N}
 = 1
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$1
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 0
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal N}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 0
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
(0
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 0
 =  {\cal X}
 (exp) {\cal N}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } SameExp(Indu)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] {\cal X}
\colon 
\forall {\cal N}
\colon 
({\cal M}
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
)
\Rightarrow 
\forall {\cal N}
\colon 
(({\cal M}
+1
)
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 =  {\cal X}
 (exp) {\cal N}
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameExp(Indu)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
({\cal M}
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal M}
+1
)
 = {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$(+1)IsPositive(N)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ({\cal M}
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$ExpPositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 = ({\cal X}
* {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$x=x+y-y
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
 = (({\cal M}
+1
)
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$A4
\mathop {\char64} (({\cal M}
+1
)
-1
)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
 = (({\cal M}
+1
)
-1
)
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
 =  {\cal X}
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$EqMultiplicationLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
* {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
)
 = ({\cal X}
* {\cal X}
 (exp) {\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$A4
\mathop {\char64} ({\cal N}
-1
)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
 = ({\cal N}
-1
)
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) 
(({\cal N}
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$PositiveToRight(Eq)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
 = ({\cal N}
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) 
(({\cal N}
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
$:}$EqMultiplicationLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
* {\cal X}
 (exp) {\cal M}
)
 = ({\cal X}
* {\cal X}
 (exp) 
(({\cal N}
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}$SubLessRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
$:}$ExpPositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) {\cal N}
 = ({\cal X}
* {\cal X}
 (exp) 
(({\cal N}
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
* {\cal X}
 (exp) 
(({\cal N}
-1
)
)
)
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity5
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
({\cal M}
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
)
\Rightarrow ({\cal M}
+1
)
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
({\cal M}
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal M}
+1
)
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
(({\cal M}
+1
)
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 =  {\cal X}
 (exp) {\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
({\cal M}
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
)
\Rightarrow 
\forall {\cal N}
\colon 
(({\cal M}
+1
)
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 =  {\cal X}
 (exp) {\cal N}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } SameExp
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] {\cal X}
\colon {\cal M}
 = {\cal N}
\vdash  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameExp
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
 = {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$SameExp(Base)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
(0
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 0
 =  {\cal X}
 (exp) {\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$SameExp(Indu)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
({\cal M}
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
)
\Rightarrow 
\forall {\cal N}
\colon 
(({\cal M}
+1
)
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 =  {\cal X}
 (exp) {\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Induction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
({\cal M}
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$A4
\mathop {\char64} {\cal N}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
 = {\cal N}
\Rightarrow  {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) {\cal N}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

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

$
[ SystemQ
\mathbf{\ lemma\ } Exp(+1)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal X}
\colon  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 = ({\cal X}
* {\cal X}
 (exp) {\cal M}
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } Exp(+1)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$(+1)IsPositive(N)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ({\cal M}
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$ExpPositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 = ({\cal X}
* {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$x=x+y-y
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
 = (({\cal M}
+1
)
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SameExp
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) {\cal M}
 =  {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$EqMultiplicationLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
* {\cal X}
 (exp) {\cal M}
)
 = ({\cal X}
* {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
* {\cal X}
 (exp) 
((({\cal M}
+1
)
-1
)
)
)
 = ({\cal X}
* {\cal X}
 (exp) {\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 = ({\cal X}
* {\cal X}
 (exp) {\cal M}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $





$
[ SystemQ
\mathbf{\ lemma\ } DistributionOut(Minus)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
\colon (({\cal X}
*{\cal Y}
)
-({\cal X}
*{\cal Z}
)
)
 = ({\cal X}
*
(({\cal Y}
-{\cal Z}
)
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } DistributionOut(Minus)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Times(-1)Left
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((-1)
*
(({\cal X}
*{\cal Z}
)
)
)
 = (-u
(({\cal X}
*{\cal Z}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
(({\cal X}
*{\cal Z}
)
)
)
 = ((-1)
*
(({\cal X}
*{\cal Z}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$timesCommutativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((-1)
*
(({\cal X}
*{\cal Z}
)
)
)
 = (({\cal X}
*{\cal Z}
)
*(-1)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$timesAssociativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal X}
*{\cal Z}
)
*(-1)
)
 = ({\cal X}
*
(({\cal Z}
*(-1)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Times(-1)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal Z}
*(-1)
)
 = (-u{\cal Z}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$EqMultiplicationLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
*
(({\cal Z}
*(-1)
)
)
)
 = ({\cal X}
*
((-u{\cal Z}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$eqTransitivity5
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
(({\cal X}
*{\cal Z}
)
)
)
 = ({\cal X}
*
((-u{\cal Z}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal X}
*{\cal Y}
)
-({\cal X}
*{\cal Z}
)
)
 = (({\cal X}
*{\cal Y}
)
+({\cal X}
*
((-u{\cal Z}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$DistributionOut
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal X}
*{\cal Y}
)
+({\cal X}
*
((-u{\cal Z}
)
)
)
)
 = ({\cal X}
*
(({\cal Y}
-{\cal Z}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal X}
*{\cal Y}
)
-({\cal X}
*{\cal Z}
)
)
 = ({\cal X}
*
(({\cal Y}
-{\cal Z}
)
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } (1/2)(x+y)-x=(1/2)(y-x)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon ((1/2
*
(({\cal X}
+{\cal Y}
)
)
)
-{\cal X}
)
 = (1/2
*
(({\cal Y}
-{\cal X}
)
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } (1/2)(x+y)-x=(1/2)(y-x)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Distribution
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(1/2
*
(({\cal X}
+{\cal Y}
)
)
)
 = ((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$eqAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*
(({\cal X}
+{\cal Y}
)
)
)
-{\cal X}
)
 = (((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
-{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$plusCommutativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
 = ((1/2
*{\cal Y}
)
+(1/2
*{\cal X}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$eqAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
-{\cal X}
)
 = (((1/2
*{\cal Y}
)
+(1/2
*{\cal X}
)
)
-{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$plusAssociativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(((1/2
*{\cal Y}
)
+(1/2
*{\cal X}
)
)
-{\cal X}
)
 = ((1/2
*{\cal Y}
)
+
(((1/2
*{\cal X}
)
-{\cal X}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$TwoHalves
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*{\cal X}
)
+(1/2
*{\cal X}
)
)
 = {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$PositiveToRight(Eq)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(1/2
*{\cal X}
)
 = ({\cal X}
-(1/2
*{\cal X}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$EqNegated
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
((1/2
*{\cal X}
)
)
)
 = (-u
(({\cal X}
-(1/2
*{\cal X}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$MinusNegated
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
(({\cal X}
-(1/2
*{\cal X}
)
)
)
)
 = ((1/2
*{\cal X}
)
-{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
((1/2
*{\cal X}
)
)
)
 = ((1/2
*{\cal X}
)
-{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*{\cal X}
)
-{\cal X}
)
 = (-u
((1/2
*{\cal X}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*{\cal Y}
)
+
(((1/2
*{\cal X}
)
-{\cal X}
)
)
)
 = ((1/2
*{\cal Y}
)
-(1/2
*{\cal X}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$DistributionOut(Minus)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*{\cal Y}
)
-(1/2
*{\cal X}
)
)
 = (1/2
*
(({\cal Y}
-{\cal X}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity6
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*
(({\cal X}
+{\cal Y}
)
)
)
-{\cal X}
)
 = (1/2
*
(({\cal Y}
-{\cal X}
)
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } y-(1/2)(x+y)=(1/2)(y-x)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon ({\cal Y}
-(1/2
*
(({\cal X}
+{\cal Y}
)
)
)
)
 = (1/2
*
(({\cal Y}
-{\cal X}
)
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } y-(1/2)(x+y)=(1/2)(y-x)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Distribution
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(1/2
*
(({\cal X}
+{\cal Y}
)
)
)
 = ((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqNegated
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
((1/2
*
(({\cal X}
+{\cal Y}
)
)
)
)
)
 = (-u
(((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal Y}
-(1/2
*
(({\cal X}
+{\cal Y}
)
)
)
)
 = ({\cal Y}
-
(((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$plusCommutativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
 = ((1/2
*{\cal Y}
)
+(1/2
*{\cal X}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$EqNegated
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
(((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
)
)
 = (-u
(((1/2
*{\cal Y}
)
+(1/2
*{\cal X}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$-x-y=-(x+y)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((-u
((1/2
*{\cal Y}
)
)
)
-(1/2
*{\cal X}
)
)
 = (-u
(((1/2
*{\cal Y}
)
+(1/2
*{\cal X}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
(((1/2
*{\cal Y}
)
+(1/2
*{\cal X}
)
)
)
)
 = ((-u
((1/2
*{\cal Y}
)
)
)
-(1/2
*{\cal X}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-u
(((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
)
)
 = ((-u
((1/2
*{\cal Y}
)
)
)
-(1/2
*{\cal X}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal Y}
-
(((1/2
*{\cal X}
)
+(1/2
*{\cal Y}
)
)
)
)
 = ({\cal Y}
+
(((-u
((1/2
*{\cal Y}
)
)
)
-(1/2
*{\cal X}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
$:}$plusAssociativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal Y}
-(1/2
*{\cal Y}
)
)
-(1/2
*{\cal X}
)
)
 = ({\cal Y}
+
(((-u
((1/2
*{\cal Y}
)
)
)
-(1/2
*{\cal X}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_I \else
\if \relax \csname lgwellbigi\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigi {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigi \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal Y}
+
(((-u
((1/2
*{\cal Y}
)
)
)
-(1/2
*{\cal X}
)
)
)
)
 = (({\cal Y}
-(1/2
*{\cal Y}
)
)
-(1/2
*{\cal X}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$TwoHalves
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*{\cal Y}
)
+(1/2
*{\cal Y}
)
)
 = {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$PositiveToRight(Eq)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(1/2
*{\cal Y}
)
 = ({\cal Y}
-(1/2
*{\cal Y}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal Y}
-(1/2
*{\cal Y}
)
)
 = (1/2
*{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$eqAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal Y}
-(1/2
*{\cal Y}
)
)
-(1/2
*{\cal X}
)
)
 = ((1/2
*{\cal Y}
)
-(1/2
*{\cal X}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
$:}$DistributionOut(Minus)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
*{\cal Y}
)
-(1/2
*{\cal X}
)
)
 = (1/2
*
(({\cal Y}
-{\cal X}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity6
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_J \else
\if \relax \csname lgwellbigj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal Y}
-(1/2
*
(({\cal X}
+{\cal Y}
)
)
)
)
 = (1/2
*
(({\cal Y}
-{\cal X}
)
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




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

$
[ SystemQ
\mathbf{\ lemma\ } PositiveBase(Base)
\colon 
\Pi {\cal X}
\colon 0
 <  {\cal X}
 (exp) 0
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveBase(Base)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$ExpZero(Exact) 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 0
 = 1
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$1
 =  {\cal X}
 (exp) 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$0<1
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < 1
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLessRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <  {\cal X}
 (exp) 0
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } PositiveBase(Indu)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal X}
\colon 0
 < {\cal X}
\vdash 0
 <  {\cal X}
 (exp) {\cal M}
\Rightarrow 0
 <  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveBase(Indu)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <  {\cal X}
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Exp(+1)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ {\cal X}
 (exp) 
(({\cal M}
+1
)
)
 = ({\cal X}
* {\cal X}
 (exp) {\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
* {\cal X}
 (exp) {\cal M}
)
 =  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$PositiveFactors
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ({\cal X}
* {\cal X}
 (exp) {\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLessRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < {\cal X}
\Rightarrow 0
 <  {\cal X}
 (exp) {\cal M}
\Rightarrow 0
 <  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <  {\cal X}
 (exp) {\cal M}
\Rightarrow 0
 <  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } PositiveBase
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal X}
\colon 0
 < {\cal X}
\vdash 0
 <  {\cal X}
 (exp) {\cal M}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveBase
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PositiveBase(Base)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <  {\cal X}
 (exp) 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PositiveBase(Indu)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <  {\cal X}
 (exp) {\cal M}
\Rightarrow 0
 <  {\cal X}
 (exp) 
(({\cal M}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Induction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <  {\cal X}
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } PositiveToRight(Eq)(1 term)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal X}
 = {\cal Y}
\vdash 0
 = ({\cal Y}
-{\cal X}
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveToRight(Eq)(1 term)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 = {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 = {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PositiveToLeft(Eq)(1 term)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal Y}
-{\cal X}
)
 = 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = ({\cal Y}
-{\cal X}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

---------

$
[ SystemQ
\mathbf{\ lemma\ } BSzero(Exact)
\colon 
\Pi {\cal M}
\colon BS({\cal M}
,0
)
 =  1/2
 (exp) {\cal M}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } BSzero(Exact)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$eqReflexivity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$BSzero
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,0
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




$
[ SystemQ
\mathbf{\ lemma\ } SameBS(2)(Base)
\colon 
\Pi {\cal M}
,\linebreak [0] N_{2} 
\colon 
\forall N_{2} 
\colon 
(0
 = N_{2} 
\Rightarrow BS({\cal M}
,0
)
 = BS({\cal M}
,N_{2} 
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameBS(2)(Base)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{2} 
 = 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$BSzero
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,N_{2} 
)
 =  1/2
 (exp) 
({\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
({\cal M}
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$BSzero(Exact)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,0
)
 =  1/2
 (exp) 
({\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,0
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = N_{2} 
\Rightarrow BS({\cal M}
,0
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
(0
 = N_{2} 
\Rightarrow BS({\cal M}
,0
)
 = BS({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } SameBS(2)(Indu)
\colon 
\Pi {\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
\colon 
\forall N_{2} 
\colon 
(N_{1} 
 = N_{2} 
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
)
\Rightarrow 
\forall N_{2} 
\colon 
((N_{1} 
+1
)
 = N_{2} 
\Rightarrow BS({\cal M}
,(N_{1} 
+1
)
)
 = BS({\cal M}
,N_{2} 
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameBS(2)(Indu)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
(N_{1} 
 = N_{2} 
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$(N_{1} 
+1
)
 = N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$(+1)IsPositive(N)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < (N_{1} 
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$BSpositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,(N_{1} 
+1
)
)
 = ( 1/2
 (exp) 
(({\cal M}
+
((N_{1} 
+1
)
)
)
)
+BS({\cal M}
,((N_{1} 
+1
)
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$x=x+y-y
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = ((N_{1} 
+1
)
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$A4
\mathop {\char64} ((N_{1} 
+1
)
-1
)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = ((N_{1} 
+1
)
-1
)
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,((N_{1} 
+1
)
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,((N_{1} 
+1
)
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,((N_{1} 
+1
)
-1
)
)
 = BS({\cal M}
,N_{1} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
(({\cal M}
+
((N_{1} 
+1
)
)
)
)
+BS({\cal M}
,((N_{1} 
+1
)
-1
)
)
)
 = ( 1/2
 (exp) 
(({\cal M}
+
((N_{1} 
+1
)
)
)
)
+BS({\cal M}
,N_{1} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$PositiveToRight(Eq)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = (N_{2} 
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$A4
\mathop {\char64} (N_{2} 
-1
)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = (N_{2} 
-1
)
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,(N_{2} 
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,(N_{2} 
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
(({\cal M}
+
((N_{1} 
+1
)
)
)
)
+BS({\cal M}
,N_{1} 
)
)
 = ( 1/2
 (exp) 
(({\cal M}
+
((N_{1} 
+1
)
)
)
)
+BS({\cal M}
,(N_{2} 
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal M}
+
((N_{1} 
+1
)
)
)
 = ({\cal M}
+N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
$:}$SameExp
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
(({\cal M}
+
((N_{1} 
+1
)
)
)
)
 =  1/2
 (exp) 
(({\cal M}
+N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
$:}$eqAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
(({\cal M}
+
((N_{1} 
+1
)
)
)
)
+BS({\cal M}
,(N_{2} 
-1
)
)
)
 = ( 1/2
 (exp) 
(({\cal M}
+N_{2} 
)
)
+BS({\cal M}
,(N_{2} 
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}$SubLessRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
$:}$BSpositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,N_{2} 
)
 = ( 1/2
 (exp) 
(({\cal M}
+N_{2} 
)
)
+BS({\cal M}
,(N_{2} 
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
(({\cal M}
+N_{2} 
)
)
+BS({\cal M}
,(N_{2} 
-1
)
)
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity6
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,(N_{1} 
+1
)
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
(N_{1} 
 = N_{2} 
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
)
\Rightarrow (N_{1} 
+1
)
 = N_{2} 
\Rightarrow BS({\cal M}
,(N_{1} 
+1
)
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
(N_{1} 
 = N_{2} 
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(N_{1} 
+1
)
 = N_{2} 
\Rightarrow BS({\cal M}
,(N_{1} 
+1
)
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
((N_{1} 
+1
)
 = N_{2} 
\Rightarrow BS({\cal M}
,(N_{1} 
+1
)
)
 = BS({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
(N_{1} 
 = N_{2} 
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
)
\Rightarrow 
\forall N_{2} 
\colon 
((N_{1} 
+1
)
 = N_{2} 
\Rightarrow BS({\cal M}
,(N_{1} 
+1
)
)
 = BS({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } SameBS(2)
\colon 
\Pi {\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
\colon N_{1} 
 = N_{2} 
\vdash BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameBS(2)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$SameBS(2)(Base)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
(0
 = N_{2} 
\Rightarrow BS({\cal M}
,0
)
 = BS({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SameBS(2)(Indu)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
(N_{1} 
 = N_{2} 
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
)
\Rightarrow 
\forall N_{2} 
\colon 
((N_{1} 
+1
)
 = N_{2} 
\Rightarrow BS({\cal M}
,(N_{1} 
+1
)
)
 = BS({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Induction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon 
(N_{1} 
 = N_{2} 
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$A4
\mathop {\char64} N_{2} 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = N_{2} 
\Rightarrow BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,N_{1} 
)
 = BS({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

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


$
[ SystemQ
\mathbf{\ lemma\ } NegativeToLeft(Less)(1 term)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon 0
 < ({\cal X}
-{\cal Y}
)
\vdash {\cal Y}
 < {\cal X}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NegativeToLeft(Less)(1 term)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ({\cal X}
-{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$LessAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(0
+{\cal Y}
)
 < (({\cal X}
-{\cal Y}
)
+{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$plus0Left
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(0
+{\cal Y}
)
 = {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SubLessLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 < (({\cal X}
-{\cal Y}
)
+{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Three2threeTerms
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal X}
-{\cal Y}
)
+{\cal Y}
)
 = (({\cal X}
+{\cal Y}
)
-{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$x=x+y-y
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 = (({\cal X}
+{\cal Y}
)
-{\cal Y}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal X}
+{\cal Y}
)
-{\cal Y}
)
 = {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal X}
-{\cal Y}
)
+{\cal Y}
)
 = {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLessRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Y}
 < {\cal X}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } BS(+1)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
\colon BS({\cal M}
,({\cal N}
+1
)
)
 = ( 1/2
 (exp) 
((({\cal M}
+{\cal N}
)
+1
)
)
+BS({\cal M}
,{\cal N}
)
)
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } BS(+1)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$(+1)IsPositive(N)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ({\cal N}
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$BSpositive
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,({\cal N}
+1
)
)
 = ( 1/2
 (exp) 
(({\cal M}
+
(({\cal N}
+1
)
)
)
)
+BS({\cal M}
,(({\cal N}
+1
)
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$plusAssociativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal M}
+{\cal N}
)
+1
)
 = ({\cal M}
+
(({\cal N}
+1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SameExp
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
((({\cal M}
+{\cal N}
)
+1
)
)
 =  1/2
 (exp) 
(({\cal M}
+
(({\cal N}
+1
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
(({\cal M}
+
(({\cal N}
+1
)
)
)
)
 =  1/2
 (exp) 
((({\cal M}
+{\cal N}
)
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$x=x+y-y
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal N}
 = (({\cal N}
+1
)
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SameBS(2)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,{\cal N}
)
 = BS({\cal M}
,(({\cal N}
+1
)
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,(({\cal N}
+1
)
-1
)
)
 = BS({\cal M}
,{\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$AddEquations
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
(({\cal M}
+
(({\cal N}
+1
)
)
)
)
+BS({\cal M}
,(({\cal N}
+1
)
-1
)
)
)
 = ( 1/2
 (exp) 
((({\cal M}
+{\cal N}
)
+1
)
)
+BS({\cal M}
,{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS({\cal M}
,({\cal N}
+1
)
)
 = ( 1/2
 (exp) 
((({\cal M}
+{\cal N}
)
+1
)
)
+BS({\cal M}
,{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } BSbound(Exact)(Base)
\colon 
\Pi {\cal M}
\colon (BS(({\cal M}
+1
)
,0
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+0
)
)
)
 =  1/2
 (exp) {\cal M}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } BSbound(Exact)(Base)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$BSzero(Exact)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS(({\cal M}
+1
)
,0
)
 =  1/2
 (exp) 
(({\cal M}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Exp(+1)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
(({\cal M}
+1
)
)
 = (1/2
* 1/2
 (exp) {\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS(({\cal M}
+1
)
,0
)
 = (1/2
* 1/2
 (exp) {\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$plus0
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal M}
+1
)
+0
)
 = ({\cal M}
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$SameExp
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
((({\cal M}
+1
)
+0
)
)
 =  1/2
 (exp) 
(({\cal M}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
((({\cal M}
+1
)
+0
)
)
 = (1/2
* 1/2
 (exp) {\cal M}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$AddEquations
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,0
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+0
)
)
)
 = ((1/2
* 1/2
 (exp) {\cal M}
)
+(1/2
* 1/2
 (exp) {\cal M}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$TwoHalves
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
* 1/2
 (exp) {\cal M}
)
+(1/2
* 1/2
 (exp) {\cal M}
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,0
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+0
)
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } BSbound(Exact)(Indu)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
\colon (BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
 =  1/2
 (exp) {\cal M}
\Rightarrow (BS(({\cal M}
+1
)
,({\cal N}
+1
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 =  1/2
 (exp) {\cal M}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } BSbound(Exact)(Indu)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$BS(+1)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS(({\cal M}
+1
)
,({\cal N}
+1
)
)
 = ( 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
+BS(({\cal M}
+1
)
,{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$eqAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,({\cal N}
+1
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 = (( 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
+BS(({\cal M}
+1
)
,{\cal N}
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$plusCommutativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
+BS(({\cal M}
+1
)
,{\cal N}
)
)
 = (BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$eqAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(( 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
+BS(({\cal M}
+1
)
,{\cal N}
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 = ((BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Exp(+1)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
 = (1/2
* 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$AddEquations
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 = ((1/2
* 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
+(1/2
* 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$TwoHalves
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((1/2
* 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
+(1/2
* 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
)
 =  1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 =  1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$Three2twoTerms
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 = (BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity5
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,({\cal N}
+1
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
 =  1/2
 (exp) {\cal M}
\Rightarrow (BS(({\cal M}
+1
)
,({\cal N}
+1
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } BSbound(Exact)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
\colon (BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
 =  1/2
 (exp) {\cal M}
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } BSbound(Exact)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$BSbound(Exact)(Base)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,0
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+0
)
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$BSbound(Exact)(Indu)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
 =  1/2
 (exp) {\cal M}
\Rightarrow (BS(({\cal M}
+1
)
,({\cal N}
+1
)
)
+ 1/2
 (exp) 
(((({\cal M}
+1
)
+{\cal N}
)
+1
)
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Induction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } BSbound
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
\colon BS(({\cal M}
+1
)
,{\cal N}
)
 <  1/2
 (exp) {\cal M}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } BSbound
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$BSbound(Exact)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$plusCommutativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
+BS(({\cal M}
+1
)
,{\cal N}
)
)
 = (BS(({\cal M}
+1
)
,{\cal N}
)
+ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$( 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
+BS(({\cal M}
+1
)
,{\cal N}
)
)
 =  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$PositiveToRight(Eq)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$ 1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
 = ( 1/2
 (exp) {\cal M}
-BS(({\cal M}
+1
)
,{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$0<1/2
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < 1/2
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$PositiveBase
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 <  1/2
 (exp) 
((({\cal M}
+1
)
+{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SubLessRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ( 1/2
 (exp) {\cal M}
-BS(({\cal M}
+1
)
,{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$NegativeToLeft(Less)(1 term)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$BS(({\cal M}
+1
)
,{\cal N}
)
 <  1/2
 (exp) {\cal M}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

---------


$
[ SystemQ
\mathbf{\ lemma\ } SameSeries(NumDiff)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] {\cal O}
,\linebreak [0] {\cal P}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
\colon {\cal O}
 = {\cal P}
\vdash N_{1} 
 = N_{2} 
\vdash |(FX
[{\cal O}
]
-FY
[N_{1} 
]
)
|
 = |(FX
[{\cal P}
]
-FY
[N_{2} 
]
)
|
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameSeries(NumDiff)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] {\cal O}
,\linebreak [0] {\cal P}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal O}
 = {\cal P}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SameSeries
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
[{\cal O}
]
 = FX
[{\cal P}
]
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SameSeries
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FY
[N_{1} 
]
 = FY
[N_{2} 
]
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$EqNegated
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(-uFY
[N_{1} 
]
)
 = (-uFY
[N_{2} 
]
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$AddEquations
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(FX
[{\cal O}
]
-FY
[N_{1} 
]
)
 = (FX
[{\cal P}
]
-FY
[N_{2} 
]
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SameNumerical
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(FX
[{\cal O}
]
-FY
[N_{1} 
]
)
|
 = |(FX
[{\cal P}
]
-FY
[N_{2} 
]
)
|
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } UStelescope(Zero)(Exact)
\colon 
\Pi {\cal M}
\colon UStelescope({\cal M}
,0
)
 = |(us
[{\cal M}
]
-us
[({\cal M}
+1
)
]
)
|
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } UStelescope(Zero)(Exact)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$eqReflexivity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$UStelescope(Zero)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,0
)
 = |(us
[{\cal M}
]
-us
[({\cal M}
+1
)
]
)
|
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } SameTelescope(2)(Base)
\colon 
\Pi {\cal M}
,\linebreak [0] N_{2} 
\colon 
\forall N_{2} 
\colon (0
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,0
)
 = UStelescope({\cal M}
,N_{2} 
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameTelescope(2)(Base)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{2} 
 = 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$UStelescope(Zero)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,N_{2} 
)
 = |(us
[{\cal M}
]
-us
[({\cal M}
+1
)
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+1
)
]
)
|
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$UStelescope(Zero)(Exact)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,0
)
 = |(us
[{\cal M}
]
-us
[({\cal M}
+1
)
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,0
)
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,0
)
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon (0
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,0
)
 = UStelescope({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } SameTelescope(2)(Indu)
\colon 
\Pi {\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
\colon 
\forall N_{2} 
\colon (N_{1} 
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
)
\Rightarrow 
\forall N_{2} 
\colon ((N_{1} 
+1
)
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,(N_{1} 
+1
)
)
 = UStelescope({\cal M}
,N_{2} 
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameTelescope(2)(Indu)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon (N_{1} 
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$(N_{1} 
+1
)
 = N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$(+1)IsPositive(N)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < (N_{1} 
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$UStelescope(Positive)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,(N_{1} 
+1
)
)
 = (|(us
[({\cal M}
+((N_{1} 
+1
)
)
)
]
-us
[({\cal M}
+(((N_{1} 
+1
)
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,((N_{1} 
+1
)
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$x=x+y-y
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = ((N_{1} 
+1
)
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$A4
\mathop {\char64} ((N_{1} 
+1
)
-1
)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = ((N_{1} 
+1
)
-1
)
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,((N_{1} 
+1
)
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,((N_{1} 
+1
)
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,((N_{1} 
+1
)
-1
)
)
 = UStelescope({\cal M}
,N_{1} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$PositiveToRight(Eq)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = (N_{2} 
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$A4
\mathop {\char64} (N_{2} 
-1
)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = (N_{2} 
-1
)
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,(N_{2} 
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,(N_{2} 
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,((N_{1} 
+1
)
-1
)
)
 = UStelescope({\cal M}
,(N_{2} 
-1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal M}
+((N_{1} 
+1
)
)
)
 = ({\cal M}
+N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_M \else
\if \relax \csname lgwellbigm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigm \fi 
$:}$eqAddition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$((N_{1} 
+1
)
+1
)
 = (N_{2} 
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_M \else
\if \relax \csname lgwellbigm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigm \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal M}
+(((N_{1} 
+1
)
+1
)
)
)
 = ({\cal M}
+((N_{2} 
+1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
$:}$SameSeries(NumDiff)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[({\cal M}
+((N_{1} 
+1
)
)
)
]
-us
[({\cal M}
+(((N_{1} 
+1
)
+1
)
)
)
]
)
|
 = |(us
[({\cal M}
+N_{2} 
)
]
-us
[({\cal M}
+((N_{2} 
+1
)
)
)
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
$:}$AddEquations
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(|(us
[({\cal M}
+((N_{1} 
+1
)
)
)
]
-us
[({\cal M}
+(((N_{1} 
+1
)
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,((N_{1} 
+1
)
-1
)
)
)
 = (|(us
[({\cal M}
+N_{2} 
)
]
-us
[({\cal M}
+((N_{2} 
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,(N_{2} 
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}$SubLessRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
$:}$UStelescope(Positive)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,N_{2} 
)
 = (|(us
[({\cal M}
+N_{2} 
)
]
-us
[({\cal M}
+((N_{2} 
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,(N_{2} 
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_r \else
\if \relax \csname lgwellr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellr \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(|(us
[({\cal M}
+N_{2} 
)
]
-us
[({\cal M}
+((N_{2} 
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,(N_{2} 
-1
)
)
)
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellp \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_s \else
\if \relax \csname lgwells\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwells {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwells \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,(N_{1} 
+1
)
)
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon (N_{1} 
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
)
\Rightarrow (N_{1} 
+1
)
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,(N_{1} 
+1
)
)
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon (N_{1} 
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(N_{1} 
+1
)
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,(N_{1} 
+1
)
)
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon ((N_{1} 
+1
)
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,(N_{1} 
+1
)
)
 = UStelescope({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon (N_{1} 
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
)
\Rightarrow 
\forall N_{2} 
\colon ((N_{1} 
+1
)
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,(N_{1} 
+1
)
)
 = UStelescope({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } SameTelescope(2)
\colon 
\Pi {\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
\colon N_{1} 
 = N_{2} 
\vdash UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameTelescope(2)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] N_{1} 
,\linebreak [0] N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = N_{2} 
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$SameTelescope(2)(Base)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon (0
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,0
)
 = UStelescope({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$SameTelescope(2)(Indu)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon (N_{1} 
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
)
\Rightarrow 
\forall N_{2} 
\colon ((N_{1} 
+1
)
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,(N_{1} 
+1
)
)
 = UStelescope({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Induction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall N_{2} 
\colon (N_{1} 
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$A4
\mathop {\char64} N_{2} 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$N_{1} 
 = N_{2} 
\Rightarrow UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,N_{1} 
)
 = UStelescope({\cal M}
,N_{2} 
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } TelescopeNumerical(Base)
\colon 
\Pi {\cal M}
\colon |(us
[{\cal M}
]
-us
[({\cal M}
+((0
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,0
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } TelescopeNumerical(Base)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$eqReflexivity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 = 0
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$UStelescope(Zero)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,0
)
 = |(us
[{\cal M}
]
-us
[({\cal M}
+1
)
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$eqReflexivity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
 = {\cal M}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$plus0Left
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(0
+1
)
 = 1
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal M}
+((0
+1
)
)
)
 = ({\cal M}
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal M}
+1
)
 = ({\cal M}
+((0
+1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SameSeries(NumDiff)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+1
)
]
)
|
 = |(us
[{\cal M}
]
-us
[({\cal M}
+((0
+1
)
)
)
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,0
)
 = |(us
[{\cal M}
]
-us
[({\cal M}
+((0
+1
)
)
)
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+((0
+1
)
)
)
]
)
|
 = UStelescope({\cal M}
,0
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqLeq
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+((0
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,0
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } TelescopeNumerical(Indu)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
\colon |(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,{\cal N}
)
\Rightarrow |(us
[{\cal M}
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,({\cal N}
+1
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } TelescopeNumerical(Indu)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,{\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$(+1)IsPositive(N)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ({\cal N}
+1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$UStelescope(Positive)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,({\cal N}
+1
)
)
 = (|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,(({\cal N}
+1
)
-1
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$x=x+y-y
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal N}
 = (({\cal N}
+1
)
-1
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(({\cal N}
+1
)
-1
)
 = {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$SameTelescope(2)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,(({\cal N}
+1
)
-1
)
)
 = UStelescope({\cal M}
,{\cal N}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$EqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,(({\cal N}
+1
)
-1
)
)
)
 = (|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$UStelescope({\cal M}
,({\cal N}
+1
)
)
 = (|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,{\cal N}
)
)
 = UStelescope({\cal M}
,({\cal N}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$LeqAdditionLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
)
 <= (|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+UStelescope({\cal M}
,{\cal N}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$subLeqRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
)
 <= UStelescope({\cal M}
,({\cal N}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$plusCommutativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
+|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
)
 = (|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
+|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$subLeqLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
+|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
)
 <= UStelescope({\cal M}
,({\cal N}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$insertMiddleTerm(Numerical)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
 <= (|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
+|(us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$leqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,({\cal N}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,{\cal N}
)
\Rightarrow |(us
[{\cal M}
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,({\cal N}
+1
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } TelescopeNumerical
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
\colon |(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,{\cal N}
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } TelescopeNumerical
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$TelescopeNumerical(Base)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+((0
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,0
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$TelescopeNumerical(Indu)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,{\cal N}
)
\Rightarrow |(us
[{\cal M}
]
-us
[({\cal M}
+((({\cal N}
+1
)
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,({\cal N}
+1
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Induction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(us
[{\cal M}
]
-us
[({\cal M}
+(({\cal N}
+1
)
)
)
]
)
|
 <= UStelescope({\cal M}
,{\cal N}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



---------------(21.10.06)

$
[ SystemQ
\mathbf{\ lemma\ } EqAdditionLeft(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
 == R(FY
)
\vdash R(FZ
)
++R(FX
)
 == R(FZ
)
++R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } EqAdditionLeft(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
++R(FX
)
 == R(FX
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FZ
)
 == R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
++R(FX
)
 == R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } x=x+(y-y)(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 = R(FX
)
++(R(FY
)
++(--R(FY
)
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } x=x+(y-y)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Plus0(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++00
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Negative(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++(--R(FY
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 == R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$EqAdditionLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++00
 == R(FX
)
++(R(FY
)
++(--R(FY
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++(R(FY
)
++(--R(FY
)
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



$
[ SystemQ
\mathbf{\ lemma\ } x=x+y-y(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 = R(FX
)
++R(FY
)
++(--R(FY
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } x=x+y-y(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$x=x+(y-y)(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++(R(FY
)
++(--R(FY
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PlusAssociativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++(--R(FY
)
)
 == R(FX
)
++(R(FY
)
++(--R(FY
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(R(FY
)
++(--R(FY
)
)
)
 == R(FX
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



----------(22.10.06)


$
[ SystemQ
\mathbf{\ lemma\ } Three2twoTerms(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
,\linebreak [0] FU
\colon R(FY
)
++R(FZ
)
 == R(FU
)
\vdash R(FX
)
++R(FY
)
++R(FZ
)
 == R(FX
)
++R(FU
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } Three2twoTerms(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
,\linebreak [0] FU
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FZ
)
 == R(FU
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqAdditionLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(R(FY
)
++R(FZ
)
)
 == R(FX
)
++R(FU
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PlusAssociativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++R(FZ
)
 == R(FX
)
++(R(FY
)
++R(FZ
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++R(FZ
)
 == R(FX
)
++R(FU
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } PositiveToRight(Less)(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
++R(FY
)
 << R(FZ
)
\vdash R(FX
)
 << R(FZ
)
++(--R(FY
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveToRight(Less)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
 << R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$lessAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++(--R(FY
)
)
 << R(FZ
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$x=x+y-y(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++(--R(FY
)
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLessLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FZ
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



$
[ SystemQ
\mathbf{\ lemma\ } Three2threeTerms(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
++R(FY
)
++R(FZ
)
 == R(FX
)
++R(FZ
)
++R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } Three2threeTerms(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FZ
)
 == R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Three2twoTerms(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++R(FZ
)
 == R(FX
)
++(R(FZ
)
++R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PlusAssociativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
++R(FY
)
 == R(FX
)
++(R(FZ
)
++R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(R(FZ
)
++R(FY
)
)
 == R(FX
)
++R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++R(FZ
)
 == R(FX
)
++R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


--------(22.10.06)

$
[ SystemQ
\mathbf{\ lemma\ } Plus0Left(R)
\colon 
\Pi FX
\colon 00
++R(FX
)
 == R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } Plus0Left(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Plus0(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++00
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++R(FX
)
 == R(FX
)
++00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++R(FX
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $






$
[ SystemQ
\mathbf{\ lemma\ } PositiveToRight(Eq)(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
++R(FY
)
 == R(FZ
)
\vdash R(FX
)
 == R(FZ
)
++(--R(FY
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveToRight(Eq)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
 == R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++(--R(FY
)
)
 == R(FZ
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$x=x+y-y(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FZ
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




$
[ SystemQ
\mathbf{\ lemma\ } SubtractEquations(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
,\linebreak [0] FU
\colon R(FX
)
++R(FZ
)
 == R(FY
)
++R(FU
)
\vdash R(FZ
)
 == R(FU
)
\vdash R(FX
)
 == R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SubtractEquations(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
,\linebreak [0] FU
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
++R(FU
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
 == R(FU
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$EqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
++(--R(FZ
)
)
 == R(FY
)
++R(FU
)
++(--R(FZ
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Plus0Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++R(FZ
)
 == R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++R(FZ
)
 == R(FU
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$PositiveToRight(Eq)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 == R(FU
)
++(--R(FZ
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FU
)
++(--R(FZ
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$EqAdditionLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++(R(FU
)
++(--R(FZ
)
)
)
 == R(FY
)
++00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$PlusAssociativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FU
)
++(--R(FZ
)
)
 == R(FY
)
++(R(FU
)
++(--R(FZ
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$Plus0(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++00
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FU
)
++(--R(FZ
)
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$x=x+y-y(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++R(FZ
)
++(--R(FZ
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




$
[ SystemQ
\mathbf{\ lemma\ } NeqAddition(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
 \neq R(FY
)
\vdash R(FX
)
++R(FZ
)
 \neq R(FY
)
++R(FZ
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NeqAddition(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 \neq R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}$==\!{}Reflexivity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
 == R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}$SubtractEquations(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 \neq R(FY
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 \neq R(FY
)
\Rightarrow R(FX
)
++R(FZ
)
 == R(FY
)
++R(FZ
)
\Rightarrow R(FX
)
++R(FZ
)
 \neq R(FY
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 \neq R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
++R(FZ
)
\Rightarrow R(FX
)
++R(FZ
)
 \neq R(FY
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$
\mbox{prop lemma imply negation}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 \neq R(FY
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

-----------(22.10.06)

$
[ SystemQ
\mathbf{\ lemma\ } PositiveToRight(Less)(1 term)(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 << R(FY
)
\vdash 00
 << R(FY
)
++(--R(FX
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveToRight(Less)(1 term)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Plus0Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++R(FX
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == 00
++R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SubLessLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$PositiveToRight(Less)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FY
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

-----------(22.10.06)



$
[ SystemQ
\mathbf{\ lemma\ } To!!==
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] \epsilon
,\linebreak [0] FX
,\linebreak [0] FY
\colon \dot{\neg}\,(SF(FX
,FY
)
)n
\vdash R(FX
)
 !!== R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } To!!==
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] \epsilon
,\linebreak [0] FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$From==
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] \epsilon
,\linebreak [0] FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
\Rightarrow SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(SF(FX
,FY
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



$
[ SystemQ
\mathbf{\ lemma\ } SwitchTerms(x<=y-z)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
\colon {\cal X}
 <= ({\cal Y}
-{\cal Z}
)
\vdash {\cal Z}
 <= ({\cal Y}
-{\cal X}
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SwitchTerms(x<=y-z)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
,\linebreak [0] {\cal Z}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 <= ({\cal Y}
-{\cal Z}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$negativeToLeft(Leq)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
+{\cal Z}
)
 <= {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$plusCommutativity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal X}
+{\cal Z}
)
 = ({\cal Z}
+{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$subLeqLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$({\cal Z}
+{\cal X}
)
 <= {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$PositiveToRight(Leq)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal Z}
 <= ({\cal Y}
-{\cal X}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $






$
[ SystemQ
\mathbf{\ lemma\ } LessNeq(F)(Helper)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] \epsilon
,\linebreak [0] FX
,\linebreak [0] FY
\colon 
\forall {\cal M}
\colon 0
 < \epsilon
\mathrel{\dot{\wedge}} ({\cal N}
 <= {\cal M}
\Rightarrow FX
[{\cal M}
]
 <= (FY
[{\cal M}
]
-\epsilon
)
)
\Rightarrow \dot{\neg}\,((0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal N}
\Rightarrow |(FX
[{\cal N}
]
-FY
[{\cal N}
]
)
|
 < \epsilon
)
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LessNeq(F)(Helper)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] \epsilon
,\linebreak [0] FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal M}
\colon 0
 < \epsilon
\mathrel{\dot{\wedge}} ({\cal N}
 <= {\cal M}
\Rightarrow FX
[{\cal M}
]
 <= (FY
[{\cal M}
]
-\epsilon
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$A4
\mathop {\char64} {\cal N}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < \epsilon
\mathrel{\dot{\wedge}} ({\cal N}
 <= {\cal N}
\Rightarrow FX
[{\cal N}
]
 <= (FY
[{\cal N}
]
-\epsilon
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < \epsilon
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal N}
 <= {\cal N}
\Rightarrow FX
[{\cal N}
]
 <= (FY
[{\cal N}
]
-\epsilon
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$leqReflexivity
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal N}
 <= {\cal N}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
[{\cal N}
]
 <= (FY
[{\cal N}
]
-\epsilon
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$SwitchTerms(x<=y-z)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\epsilon
 <= (FY
[{\cal N}
]
-FX
[{\cal N}
]
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$x<=|x|
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(FY
[{\cal N}
]
-FX
[{\cal N}
]
)
 <= |(FY
[{\cal N}
]
-FX
[{\cal N}
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$leqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\epsilon
 <= |(FY
[{\cal N}
]
-FX
[{\cal N}
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$NumericalDifference
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|(FY
[{\cal N}
]
-FX
[{\cal N}
]
)
|
 = |(FX
[{\cal N}
]
-FY
[{\cal N}
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$subLeqRight
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\epsilon
 <= |(FX
[{\cal N}
]
-FY
[{\cal N}
]
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$toNotLess
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(|(FX
[{\cal N}
]
-FY
[{\cal N}
]
)
|
 < \epsilon
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToNegatedDoubleImply
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,((0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal N}
\Rightarrow |(FX
[{\cal N}
]
-FY
[{\cal N}
]
)
|
 < \epsilon
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] \epsilon
,\linebreak [0] FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal M}
\colon 0
 < \epsilon
\mathrel{\dot{\wedge}} ({\cal N}
 <= {\cal M}
\Rightarrow FX
[{\cal M}
]
 <= (FY
[{\cal M}
]
-\epsilon
)
)
\Rightarrow \dot{\neg}\,((0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal N}
\Rightarrow |(FX
[{\cal N}
]
-FY
[{\cal N}
]
)
|
 < \epsilon
)
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } LessNeq(F)
\colon 
\Pi {\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] \epsilon
,\linebreak [0] FX
,\linebreak [0] FY
\colon FX
<_{f}FY
\vdash \dot{\neg}\,(SF(FX
,FY
)
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LessNeq(F)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal M}
,\linebreak [0] {\cal N}
,\linebreak [0] \epsilon
,\linebreak [0] FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_x \else
\if \relax \csname lgwellx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellx \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists (EPob)
\colon 
\exists 
\mathit{n}
\colon 
\forall 
\mathit{m}
\colon 0
 < (EPob)
\mathrel{\dot{\wedge}} (
\mathit{n}
 <= 
\mathit{m}
\Rightarrow FX
[
\mathit{m}
]
 <= (FY
[
\mathit{m}
]
-(EPob)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_x \else
\if \relax \csname lgwellx\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellx {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellx \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists \epsilon
\colon 
\exists {\cal N}
\colon 
\forall {\cal M}
\colon 0
 < \epsilon
\mathrel{\dot{\wedge}} ({\cal N}
 <= {\cal M}
\Rightarrow FX
[{\cal M}
]
 <= (FY
[{\cal M}
]
-\epsilon
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$LessNeq(F)(Helper)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal M}
\colon 0
 < \epsilon
\mathrel{\dot{\wedge}} ({\cal N}
 <= {\cal M}
\Rightarrow FX
[{\cal M}
]
 <= (FY
[{\cal M}
]
-\epsilon
)
)
\Rightarrow \dot{\neg}\,((0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal N}
\Rightarrow |(FX
[{\cal N}
]
-FY
[{\cal N}
]
)
|
 < \epsilon
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$TwiceExistMP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,((0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal N}
\Rightarrow |(FX
[{\cal N}
]
-FY
[{\cal N}
]
)
|
 < \epsilon
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$IntroExist
\mathop {\char64} {\cal N}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists {\cal M}
\colon \dot{\neg}\,((0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal M}
\Rightarrow |(FX
[{\cal M}
]
-FY
[{\cal M}
]
)
|
 < \epsilon
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall {\cal N}
\colon 
\exists {\cal M}
\colon \dot{\neg}\,((0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal M}
\Rightarrow |(FX
[{\cal M}
]
-FY
[{\cal M}
]
)
|
 < \epsilon
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$IntroExist
\mathop {\char64} \epsilon
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists \epsilon
\colon 
\forall {\cal N}
\colon 
\exists {\cal M}
\colon \dot{\neg}\,((0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal M}
\Rightarrow |(FX
[{\cal M}
]
-FY
[{\cal M}
]
)
|
 < \epsilon
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$ToNegatedAEA 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\forall \epsilon
\colon 
\exists {\cal N}
\colon 
\forall {\cal M}
\colon (0
 < \epsilon
\Rightarrow {\cal N}
 <= {\cal M}
\Rightarrow |(FX
[{\cal M}
]
-FY
[{\cal M}
]
)
|
 < \epsilon
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\forall (EPob)
\colon 
\exists 
\mathit{n}
\colon 
\forall 
\mathit{m}
\colon (0
 < (EPob)
\Rightarrow 
\mathit{n}
 <= 
\mathit{m}
\Rightarrow |(FX
[
\mathit{m}
]
-FY
[
\mathit{m}
]
)
|
 < (EPob)
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(SF(FX
,FY
)
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } LessNeq(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 << R(FY
)
\vdash R(FX
)
 !!== R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LessNeq(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$LessNeq(F)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(SF(FX
,FY
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$To!!==
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $





$
[ SystemQ
\mathbf{\ lemma\ } PositiveToRight(Less)(1 term)
\colon 
\Pi {\cal X}
,\linebreak [0] {\cal Y}
\colon {\cal X}
 < {\cal Y}
\vdash 0
 < ({\cal Y}
-{\cal X}
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveToRight(Less)(1 term)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
,\linebreak [0] {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 < {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$plus0Left
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(0
+{\cal X}
)
 = {\cal X}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$eqSymmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal X}
 = (0
+{\cal X}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SubLessLeft
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(0
+{\cal X}
)
 < {\cal Y}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$PositiveToRight(Less)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0
 < ({\cal Y}
-{\cal X}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } ToLeq(Advanced)(R)
\colon 
\Pi FEP
,\linebreak [0] FX
,\linebreak [0] FY
\colon 00
 << R(FEP
)
\Rightarrow R(FX
)
++R(FEP
)
 \neq R(FY
)
\vdash R(FY
)
 <<== R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } ToLeq(Advanced)(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FEP
,\linebreak [0] FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PositiveToRight(Less)(1 term)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FY
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(R(FY
)
++(--R(FX
)
)
)
 = R(FY
)
++(--R(FX
)
)
++R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Three2threeTerms(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++(--R(FX
)
)
++R(FX
)
 = R(FY
)
++R(FX
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$x=x+y-y(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 = R(FY
)
++R(FX
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FX
)
++(--R(FX
)
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(R(FY
)
++(--R(FX
)
)
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$JoinConjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FY
)
++(--R(FX
)
)
\mathrel{\dot{\wedge}} R(FX
)
++(R(FY
)
++(--R(FX
)
)
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$IntroExist
\mathop {\char64} R(FY
)
++(--R(FX
)
)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\exists FEP
\colon 00
 << R(FEP
)
\mathrel{\dot{\wedge}} R(FX
)
++R(FEP
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FEP
,\linebreak [0] FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
\Rightarrow 
\exists FEP
\colon 00
 << R(FEP
)
\mathrel{\dot{\wedge}} R(FX
)
++R(FEP
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FEP
)
\Rightarrow R(FX
)
++R(FEP
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$ToNegatedAnd
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,((00
 << R(FEP
)
\mathrel{\dot{\wedge}} R(FX
)
++R(FEP
)
 == R(FY
)
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Gen
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\forall FEP
\colon \dot{\neg}\,((00
 << R(FEP
)
\mathrel{\dot{\wedge}} R(FX
)
++R(FEP
)
 == R(FY
)
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$(A~)to(~E)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(
\exists FEP
\colon 00
 << R(FEP
)
\mathrel{\dot{\wedge}} R(FX
)
++R(FEP
)
 == R(FY
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FX
)
 << R(FY
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromNotLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

--------(23.10.06)


$
[ SystemQ
\mathbf{\ lemma\ } LeqNeqLess(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 <<== R(FY
)
\vdash R(FX
)
 !!== R(FY
)
\vdash R(FX
)
 << R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LeqNeqLess(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$LeqAntisymmetry(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
\Rightarrow R(FX
)
 !!== R(FY
)
\Rightarrow R(FY
)
 <<== R(FX
)
\Rightarrow \dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
\Rightarrow \dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$
\mbox{prop lemma imply negation}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$ToLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } SubLeqLeft(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
 == R(FY
)
\vdash R(FX
)
 <<== R(FZ
)
\vdash R(FY
)
 <<== R(FZ
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SubLeqLeft(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$
\mbox{lemma eqLeq(R)}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$LeqTransitivity(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FZ
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } LeqLessTransitivity(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
 <<== R(FY
)
\vdash R(FY
)
 << R(FZ
)
\vdash R(FX
)
 << R(FZ
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LeqLessTransitivity(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 << R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$LessLeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$LessNeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 !!== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$SubLeqLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$LeqAntisymmetry(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 == R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
\Rightarrow R(FY
)
 << R(FZ
)
\Rightarrow R(FX
)
 == R(FZ
)
\Rightarrow R(FX
)
 !!== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 << R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$MP2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FZ
)
\Rightarrow R(FX
)
 !!== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$
\mbox{prop lemma imply negation}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
$:}$LessLeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
$:}$LeqTransitivity(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$LeqNeqLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_q \else
\if \relax \csname lgwellq\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellq {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellq \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FZ
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

-----------(23.10.06)


$
[ SystemQ
\mathbf{\ lemma\ } NegativeToLeft(Eq)(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
 == R(FY
)
++(--R(FZ
)
)
\vdash R(FX
)
++R(FZ
)
 == R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NegativeToLeft(Eq)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
++(--R(FZ
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
++(--R(FZ
)
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Three2threeTerms(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++(--R(FZ
)
)
++R(FZ
)
 == R(FY
)
++R(FZ
)
++(--R(FZ
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$x=x+y-y(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 == R(FY
)
++R(FZ
)
++(--R(FZ
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FZ
)
++(--R(FZ
)
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




$
[ SystemQ
\mathbf{\ lemma\ } NegativeToRight(Less)(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
++(--R(FY
)
)
 << R(FZ
)
\vdash R(FX
)
 << R(FZ
)
++R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NegativeToRight(Less)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FY
)
)
 << R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$lessAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FY
)
)
++R(FY
)
 << R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Three2threeTerms(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FY
)
)
++R(FY
)
 == R(FX
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$x=x+y-y(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++(--R(FY
)
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FY
)
)
++R(FY
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLessLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } !!==Symmetry
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 !!== R(FY
)
\vdash R(FY
)
 !!== R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } !!==Symmetry
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 == R(FX
)
\Rightarrow R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MT
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 !!== R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


-------------(23.10.06)


$
[ SystemQ
\mathbf{\ lemma\ } NegativeToRight(Eq)(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
++(--R(FY
)
)
 == R(FZ
)
\vdash R(FX
)
 == R(FZ
)
++R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NegativeToRight(Eq)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FY
)
)
 == R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FY
)
)
++R(FY
)
 == R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$x=x+y-y(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Three2threeTerms(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
++(--R(FY
)
)
 == R(FX
)
++(--R(FY
)
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FZ
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } NegativeToRight(Eq)(1 term)(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
++(--R(FY
)
)
 == 00
\vdash R(FX
)
 == R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NegativeToRight(Eq)(1 term)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FY
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$NegativeToRight(Eq)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == 00
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Plus0Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++R(FY
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } DoubleMinus(R)
\colon 
\Pi FX
\colon (--(--R(FX
)
)
)
 == R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } DoubleMinus(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Negative(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
++(--(--R(FX
)
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
++(--(--R(FX
)
)
)
 == (--(--R(FX
)
)
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FX
)
)
)
++(--R(FX
)
)
 == (--R(FX
)
)
++(--(--R(FX
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FX
)
)
)
++(--R(FX
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$NegativeToRight(Eq)(1 term)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FX
)
)
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $






$
[ SystemQ
\mathbf{\ lemma\ } UniqueNegative(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
++R(FY
)
 == 00
\vdash R(FX
)
++R(FZ
)
 == 00
\vdash R(FY
)
 == R(FZ
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } UniqueNegative(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FY
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FX
)
 == R(FX
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FX
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$PositiveToRight(Eq)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 == 00
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
++R(FX
)
 == R(FX
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
++R(FX
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$PositiveToRight(Eq)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
 == 00
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++(--R(FX
)
)
 == R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 == R(FZ
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $





$
[ SystemQ
\mathbf{\ lemma\ } SubtractEquationsLeft(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
,\linebreak [0] FU
\colon R(FX
)
++R(FZ
)
 == R(FY
)
++R(FU
)
\vdash R(FX
)
 == R(FY
)
\vdash R(FZ
)
 == R(FU
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SubtractEquationsLeft(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
,\linebreak [0] FU
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
++R(FU
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
++R(FX
)
 == R(FX
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FU
)
 == R(FU
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
++R(FX
)
 == R(FU
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubtractEquations(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
 == R(FU
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } EqNegated(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 == R(FY
)
\vdash (--R(FX
)
)
 == (--R(FY
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } EqNegated(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Negative(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FX
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Negative(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++(--R(FY
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 == R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FX
)
)
 == R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubtractEquationsLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 == (--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




$
[ SystemQ
\mathbf{\ lemma\ } NeqNegated(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 !!== R(FY
)
\vdash (--R(FX
)
)
 !!== (--R(FY
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NeqNegated(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 == (--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}$EqNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FX
)
)
)
 == (--(--R(FY
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}$DoubleMinus(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FX
)
)
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == (--(--R(FX
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}$DoubleMinus(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FY
)
)
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 !!== (--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_H \else
\if \relax \csname lgwellbigh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
\Rightarrow (--R(FX
)
)
 == (--R(FY
)
)
\Rightarrow \dot{\neg}\,((--R(FX
)
)
 == (--R(FY
)
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 == (--R(FY
)
)
\Rightarrow \dot{\neg}\,((--R(FX
)
)
 == (--R(FY
)
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$
\mbox{prop lemma imply negation}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,((--R(FX
)
)
 == (--R(FY
)
)
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

-----------(23.10.06)




$
[ SystemQ
\mathbf{\ lemma\ } SubLeqRight(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
 == R(FY
)
\vdash R(FZ
)
 <<== R(FX
)
\vdash R(FZ
)
 <<== R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SubLeqRight(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$
\mbox{lemma eqLeq(R)}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$LeqTransitivity(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } LeqNegated(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 <<== R(FY
)
\vdash (--R(FY
)
)
 <<== (--R(FX
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LeqNegated(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$leqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FX
)
)
 <<== R(FY
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Negative(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FX
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$SubLeqLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FY
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++(--R(FX
)
)
 == (--R(FX
)
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$SubLeqRight(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== (--R(FX
)
)
++R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$leqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++(--R(FY
)
)
 <<== (--R(FX
)
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$Plus0Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++(--R(FY
)
)
 == (--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$x=x+y-y(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 == (--R(FX
)
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
++R(FY
)
++(--R(FY
)
)
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$SubLeqLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FY
)
)
 <<== (--R(FX
)
)
++R(FY
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLeqRight(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FY
)
)
 <<== (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $




$
[ SystemQ
\mathbf{\ lemma\ } LessNegated(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 << R(FY
)
\vdash (--R(FY
)
)
 << (--R(FX
)
)
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LessNegated(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$LessLeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$LeqNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FY
)
)
 <<== (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$LessNeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$NeqNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 !!== (--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$!!==Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FY
)
)
 !!== (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$LeqNeqLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FY
)
)
 << (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } -0=0(R)
\colon (--00
)
 == 00
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } -0=0(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Negative(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++(--00
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Plus0(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++00
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$UniqueNegative(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--00
)
 == 00
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } NegativeNegated(R)
\colon 
\Pi FX
\colon R(FX
)
 << 00
\vdash 00
 << (--R(FX
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NegativeNegated(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$LessNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--00
)
 << (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$-0=0(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--00
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLessLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } LeqTotality(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 <<== R(FY
)
 \mathrel{\dot{\vee}} R(FY
)
 <<== R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LeqTotality(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FX
)
 <<== R(FY
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$ToLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$LessLeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FX
)
 <<== R(FY
)
)n
\Rightarrow R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
 \mathrel{\dot{\vee}} R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } FromLeqGeq(R)
\colon 
\Pi {\cal A}
,\linebreak [0] FX
,\linebreak [0] FY
\colon R(FX
)
 <<== R(FY
)
\Rightarrow {\cal A}
\vdash R(FY
)
 <<== R(FX
)
\Rightarrow {\cal A}
\vdash {\cal A}
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } FromLeqGeq(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
,\linebreak [0] FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
\Rightarrow {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
\Rightarrow {\cal A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$LeqTotality(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
 \mathrel{\dot{\vee}} R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromDisjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

--------(24.10.06)

$
[ SystemQ
\mathbf{\ lemma\ } FromLess(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 << R(FY
)
\vdash \dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } FromLess(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$LessLeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$LeqAntisymmetry(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$LessNeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
\Rightarrow R(FY
)
 <<== R(FX
)
\Rightarrow \dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
\Rightarrow \dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$
\mbox{prop lemma imply negation}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FY
)
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } from<<==
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 <<== R(FY
)
\vdash FX
\leq_{f}FY
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } from<<==
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
 \mathrel{\dot{\vee}} R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
 \mathrel{\dot{\vee}} R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$WeakenOr2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
 \mathrel{\dot{\vee}} SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$From==
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$WeakenOr1
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
 \mathrel{\dot{\vee}} SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
\Rightarrow FX
<_{f}FY
 \mathrel{\dot{\vee}} SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 = R(FY
)
\Rightarrow FX
<_{f}FY
 \mathrel{\dot{\vee}} SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$FromDisjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
 \mathrel{\dot{\vee}} SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
\leq_{f}FY
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } NonnegativeNumerical(F)
\colon 
\Pi FX
\colon 0f
\leq_{f}FX
\vdash |fFX
|
 = FX
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NonnegativeNumerical(F)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0f
\leq_{f}FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$NumericalF
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(0f
\leq_{f}FX
\Rightarrow |fFX
|
 = FX
)
\mathrel{\dot{\wedge}} (\dot{\neg}\,(0f
\leq_{f}FX
)n
\Rightarrow |fFX
|
 = -_{f}FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$FirstConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0f
\leq_{f}FX
\Rightarrow |fFX
|
 = FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|fFX
|
 = FX
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } NonnegativeNumerical(R)
\colon 
\Pi FX
\colon 00
 <<== R(FX
)
\vdash |rR(FX
)
|
 = R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NonnegativeNumerical(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$from<<==
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0f
\leq_{f}FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$NonnegativeNumerical(F)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|fFX
|
 = FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$(Adgic)SameR
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(|fFX
|
)
 = R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 = R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } to<<==
\colon 
\Pi FX
,\linebreak [0] FY
\colon FX
\leq_{f}FY
\vdash R(FX
)
 <<== R(FY
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } to<<==
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
\leq_{f}FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
 \mathrel{\dot{\vee}} SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$WeakenOr2
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
 \mathrel{\dot{\vee}} R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$SF(FX
,FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$To==
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$WeakenOr1
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
 \mathrel{\dot{\vee}} R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$FX
<_{f}FY
\Rightarrow R(FX
)
 << R(FY
)
 \mathrel{\dot{\vee}} R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$SF(FX
,FY
)
\Rightarrow R(FX
)
 << R(FY
)
 \mathrel{\dot{\vee}} R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$FromDisjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
 \mathrel{\dot{\vee}} R(FX
)
 = R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



$
[ SystemQ
\mathbf{\ lemma\ } NegativeNumerical(F)
\colon 
\Pi FX
\colon \dot{\neg}\,(0f
\leq_{f}FX
)n
\vdash |fFX
|
 = -_{f}FX
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NegativeNumerical(F)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(0f
\leq_{f}FX
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$NumericalF
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(0f
\leq_{f}FX
\Rightarrow |fFX
|
 = FX
)
\mathrel{\dot{\wedge}} (\dot{\neg}\,(0f
\leq_{f}FX
)n
\Rightarrow |fFX
|
 = -_{f}FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SecondConjunct
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(0f
\leq_{f}FX
)n
\Rightarrow |fFX
|
 = -_{f}FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|fFX
|
 = -_{f}FX
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } NegativeNumerical(R)
\colon 
\Pi FX
\colon R(FX
)
 << 00
\vdash |rR(FX
)
|
 == (--R(FX
)
)
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NegativeNumerical(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$0f
\leq_{f}FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$FromLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(00
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$to<<==
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromContradiction
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(0f
\leq_{f}FX
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
\Rightarrow 0f
\leq_{f}FX
\Rightarrow \dot{\neg}\,(0f
\leq_{f}FX
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$0f
\leq_{f}FX
\Rightarrow \dot{\neg}\,(0f
\leq_{f}FX
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$
\mbox{prop lemma imply negation}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(0f
\leq_{f}FX
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$NegativeNumerical(F)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|fFX
|
 = -_{f}FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$(Adgic)SameR
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(|fFX
|
)
 = R(-_{f}FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 = (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } 0<=|x|(R)
\colon 
\Pi FX
\colon 00
 <<== |rR(FX
)
|
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } 0<=|x|(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$NonnegativeNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLeqRight(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(00
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$ToLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$NegativeNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 == |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$NegativeNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$LessLeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLeqRight(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
)
\Rightarrow 00
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(00
 <<== R(FX
)
)n
\Rightarrow 00
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromNegations
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

-----------(24.10.06)

$
[ SystemQ
\mathbf{\ lemma\ } PositiveNegated(R)
\colon 
\Pi FX
\colon 00
 << R(FX
)
\vdash (--R(FX
)
)
 << 00
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveNegated(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$LessNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 << (--00
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$-0=0(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--00
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SubLessRight(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 << 00
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } AddEquations(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
,\linebreak [0] FU
\colon R(FX
)
 == R(FY
)
\vdash R(FZ
)
 == R(FU
)
\vdash R(FX
)
++R(FZ
)
 == R(FY
)
++R(FU
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } AddEquations(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
,\linebreak [0] FU
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FZ
)
 == R(FU
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$EqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
++R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$EqAdditionLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++R(FZ
)
 == R(FY
)
++R(FU
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++R(FZ
)
 == R(FY
)
++R(FU
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } DistributionOut(R)
\colon 
\Pi FX
,\linebreak [0] FY
,\linebreak [0] FZ
\colon R(FX
)
**R(FY
)
++R(FX
)
**R(FZ
)
 == R(FX
)
**(R(FY
)
++R(FZ
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } DistributionOut(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
,\linebreak [0] FZ
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Distribution(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(R(FY
)
++R(FZ
)
)
 == R(FX
)
**R(FY
)
++R(FX
)
**R(FZ
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**R(FY
)
++R(FX
)
**R(FZ
)
 == R(FX
)
**(R(FY
)
++R(FZ
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



$
[ SystemQ
\mathbf{\ lemma\ } x*0+x=x(R)
\colon 
\Pi FX
\colon R(FX
)
**00
++R(FX
)
 == R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } x*0+x=x(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Times1(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**
\mbox{01}
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FX
)
**
\mbox{01}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqAdditionLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
++R(FX
)
 == R(FX
)
**00
++R(FX
)
**
\mbox{01}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$Distribution(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(00
++
\mbox{01}
)
 == R(FX
)
**00
++R(FX
)
**
\mbox{01}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
++R(FX
)
**
\mbox{01}
 == R(FX
)
**(00
++
\mbox{01}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Plus0Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++
\mbox{01}
 == 
\mbox{01}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$EqMultiplicationLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(00
++
\mbox{01}
)
 == R(FX
)
**
\mbox{01}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity5
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
++R(FX
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

--------(24.10.06)




$
[ SystemQ
\mathbf{\ lemma\ } x*0=0(R)(fff)
\colon 
\Pi FX
\colon R(FX
)
**00
 == 00
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } x*0=0(R)(fff)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$x=x+(y-y)(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
 == R(FX
)
**00
++(R(FX
)
++(--R(FX
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PlusAssociativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
++R(FX
)
++(--R(FX
)
)
 == R(FX
)
**00
++(R(FX
)
++(--R(FX
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
++(R(FX
)
++(--R(FX
)
)
)
 == R(FX
)
**00
++R(FX
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$x*0+x=x(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
++R(FX
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$EqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
++R(FX
)
++(--R(FX
)
)
 == R(FX
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Negative(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
++(--R(FX
)
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity5
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
 == 00
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } Times(-1)(R)
\colon 
\Pi FX
\colon R(FX
)
**(--01)
 == (--R(FX
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } Times(-1)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Negative(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\mbox{01}
++(--01)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
++
\mbox{01}
 == 
\mbox{01}
++(--01)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
++
\mbox{01}
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$EqMultiplicationLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**((--01)
++
\mbox{01}
)
 == R(FX
)
**00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$x*0=0(R)(fff)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**00
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**((--01)
++
\mbox{01}
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$Distribution(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**((--01)
++
\mbox{01}
)
 == R(FX
)
**(--01)
++R(FX
)
**
\mbox{01}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(--01)
++R(FX
)
**
\mbox{01}
 == R(FX
)
**((--01)
++
\mbox{01}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(--01)
++R(FX
)
**
\mbox{01}
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$PositiveToRight(Eq)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(--01)
 == 00
++(--(R(FX
)
**
\mbox{01}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
$:}$Plus0Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
++(--(R(FX
)
**
\mbox{01}
)
)
 == (--(R(FX
)
**
\mbox{01}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(--01)
 == (--(R(FX
)
**
\mbox{01}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
$:}$Times1(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**
\mbox{01}
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
$:}$EqNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelln \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(R(FX
)
**
\mbox{01}
)
)
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(--01)
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } Times(-1)Left(R)
\colon 
\Pi FX
\colon (--01)
**R(FX
)
 == (--R(FX
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } Times(-1)Left(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Times(-1)(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
**(--01)
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$TimesCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
**R(FX
)
 == R(FX
)
**(--01)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
**R(FX
)
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



(************)

$
[ SystemQ
\mathbf{\ lemma\ } -x-y=-(x+y)(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon (--R(FX
)
)
++(--R(FY
)
)
 == (--(R(FX
)
++R(FY
)
)
)
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } -x-y=-(x+y)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Times(-1)Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
**R(FX
)
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Times(-1)Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
**R(FY
)
 == (--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$AddEquations(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
**R(FX
)
++(--01)
**R(FY
)
 == (--R(FX
)
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
++(--R(FY
)
)
 == (--01)
**R(FX
)
++(--01)
**R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$DistributionOut(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
**R(FX
)
++(--01)
**R(FY
)
 == (--01)
**(R(FX
)
++R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Times(-1)Left(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--01)
**(R(FX
)
++R(FY
)
)
 == (--(R(FX
)
++R(FY
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
++(--R(FY
)
)
 == (--(R(FX
)
++R(FY
)
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } LessTotality(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 << R(FY
)
 \mathrel{\dot{\vee}} R(FX
)
 == R(FY
)
 \mathrel{\dot{\vee}} R(FY
)
 << R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } LessTotality(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FX
)
 << R(FY
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 !!== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$FromNotLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$!!==Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 !!== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$LeqNeqLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(R(FX
)
 << R(FY
)
)n
\Rightarrow R(FX
)
 !!== R(FY
)
\Rightarrow R(FY
)
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Repetition
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << R(FY
)
 \mathrel{\dot{\vee}} R(FX
)
 == R(FY
)
 \mathrel{\dot{\vee}} R(FY
)
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } SameNumerical(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon R(FX
)
 == R(FY
)
\vdash |rR(FX
)
|
 == |rR(FY
)
|
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SameNumerical(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
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$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}$NonnegativeNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$SubLeqRight(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$NonnegativeNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FY
)
|
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 == |rR(FY
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
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$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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\rhd 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == |rR(FY
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(00
 <<== R(FX
)
)n
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$ToLess(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$NegativeNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$SubLessLeft(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
 << 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
$:}$NegativeNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_h \else
\if \relax \csname lgwellh\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellh {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellh \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FY
)
|
 == (--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_i \else
\if \relax \csname lgwelli\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelli {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelli \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FY
)
)
 == |rR(FY
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$EqNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 == (--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_k \else
\if \relax \csname lgwellk\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellk {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellk \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_j \else
\if \relax \csname lgwellj\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellj {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellj \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == |rR(FY
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelll \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
)
\Rightarrow R(FX
)
 == R(FY
)
\Rightarrow |rR(FX
)
|
 == |rR(FY
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$Ded
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\if \relax \csname lgwprooflinep\endcsname L_l \else
\if \relax \csname lgwelll\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelll {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$\dot{\neg}\,(00
 <<== R(FX
)
)n
\Rightarrow R(FX
)
 == R(FY
)
\Rightarrow |rR(FX
)
|
 == |rR(FY
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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$:}$FromNegations
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_n \else
\if \relax \csname lgwelln\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelln {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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\rhd 
\if \relax \csname lgwprooflinep\endcsname L_o \else
\if \relax \csname lgwello\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwello {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwello \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == R(FY
)
\Rightarrow |rR(FX
)
|
 == |rR(FY
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$MP
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_p \else
\if \relax \csname lgwellp\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellp {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
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\rhd 
\if \relax \csname lgwprooflinep\endcsname L_m \else
\if \relax \csname lgwellm\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellm {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellm \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == |rR(FY
)
|
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } MinusNegated(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon (--(R(FX
)
++(--R(FY
)
)
)
)
 == R(FY
)
++(--R(FX
)
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } MinusNegated(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$DoubleMinus(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FY
)
)
)
 == R(FY
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqAddition(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FY
)
)
)
++(--R(FX
)
)
 == R(FY
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++(--R(FX
)
)
 == (--(--R(FY
)
)
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$-x-y=-(x+y)(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FY
)
)
)
++(--R(FX
)
)
 == (--((--R(FY
)
)
++R(FX
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$PlusCommutativity(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FY
)
)
++R(FX
)
 == R(FX
)
++(--R(FY
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$EqNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--((--R(FY
)
)
++R(FX
)
)
)
 == (--(R(FX
)
++(--R(FY
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FY
)
++(--R(FX
)
)
 == (--(R(FX
)
++(--R(FY
)
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(R(FX
)
++(--R(FY
)
)
)
)
 == R(FY
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


-----------(24.10.06)

$
[ SystemQ
\mathbf{\ lemma\ } PositiveNumerical(R)
\colon 
\Pi FX
\colon 00
 << R(FX
)
\vdash |rR(FX
)
|
 == R(FX
)
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } PositiveNumerical(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$LessLeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$NonnegativeNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
[ SystemQ
\mathbf{\ lemma\ } SignNumerical(+)(R)
\colon 
\Pi FX
\colon 00
 << R(FX
)
\vdash |rR(FX
)
|
 == |r(--R(FX
)
)
|
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SignNumerical(+)(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$PositiveNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$PositiveNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 << 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$NegativeNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|r(--R(FX
)
)
|
 == (--(--R(FX
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$DoubleMinus(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FX
)
)
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|r(--R(FX
)
)
|
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_g \else
\if \relax \csname lgwellg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellg \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $



$
[ SystemQ
\mathbf{\ lemma\ } SignNumerical(R)
\colon 
\Pi FX
\colon |rR(FX
)
|
 == |r(--R(FX
)
)
|
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } SignNumerical(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$NegativeNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SignNumerical(+)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|r(--R(FX
)
)
|
 == |r(--(--R(FX
)
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$DoubleMinus(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(--R(FX
)
)
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$SameNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|r(--(--R(FX
)
)
)
|
 == |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|r(--R(FX
)
)
|
 == |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$EqNegated(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 == (--00
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$-0=0(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--00
)
 == 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
$:}$eqTransitivity4
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_d \else
\if \relax \csname lgwelld\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelld {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelld \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--R(FX
)
)
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_e \else
\if \relax \csname lgwelle\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelle {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelle \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == (--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SameNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_f \else
\if \relax \csname lgwellf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$SignNumerical(+)(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
\Rightarrow |rR(FX
)
|
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == 00
\Rightarrow |rR(FX
)
|
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_C \else
\if \relax \csname lgwellbigc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 << R(FX
)
\Rightarrow |rR(FX
)
|
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
$:}$LessTotality(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 << 00
 \mathrel{\dot{\vee}} R(FX
)
 == 00
 \mathrel{\dot{\vee}} 00
 << R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$From3Disjuncts
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_G \else
\if \relax \csname lgwellbigg\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigg {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigg \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_D \else
\if \relax \csname lgwellbigd\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigd {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigd \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_E \else
\if \relax \csname lgwellbige\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbige {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbige \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_F \else
\if \relax \csname lgwellbigf\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigf {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigf \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == |r(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

$
[ SystemQ
\mathbf{\ lemma\ } NumericalDifference(R)
\colon 
\Pi FX
,\linebreak [0] FY
\colon |rR(FX
)
++(--R(FY
)
)
|
 == |rR(FY
)
++(--R(FX
)
)
|
]$

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } NumericalDifference(R)
\colon 
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
,\linebreak [0] FY
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$SignNumerical(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
++(--R(FY
)
)
|
 == |r(--(R(FX
)
++(--R(FY
)
)
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$MinusNegated(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$(--(R(FX
)
++(--R(FY
)
)
)
)
 == R(FY
)
++(--R(FX
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$SameNumerical(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|r(--(R(FX
)
++(--R(FY
)
)
)
)
|
 == |rR(FY
)
++(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$eqTransitivity
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
++(--R(FY
)
)
|
 == |rR(FY
)
++(--R(FX
)
)
|
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $

--------(25.10.06)

$
[ SystemQ
\mathbf{\ lemma\ } x<=|x|(R)
\colon 
\Pi FX
\colon R(FX
)
 <<== |rR(FX
)
|
]$


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 SystemQ
\mathbf {\ proof\ of\ } x<=|x|(R)
\colon 
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$NonnegativeNumerical(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$|rR(FX
)
|
 == R(FX
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
$:}$==\!{}Symmetry
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 == |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$
\mbox{lemma eqLeq(R)}
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_c \else
\if \relax \csname lgwellc\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellc {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellc \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$Begin
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}\makebox [0.4\textwidth ][l]{$Premise{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== 00
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$0<=|x|(R)
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$LeqTransitivity(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
$:}$Block {}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$End
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad
\parbox [t]{0.4\textwidth }{$FX
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_A \else
\if \relax \csname lgwellbiga\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbiga {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbiga \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$00
 <<== R(FX
)
\Rightarrow R(FX
)
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
$:}$Ded
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_B \else
\if \relax \csname lgwellbigb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== 00
\Rightarrow R(FX
)
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$FromLeqGeq(R)
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_a \else
\if \relax \csname lgwella\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwella {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwella \fi 
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_b \else
\if \relax \csname lgwellb\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellb {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellb \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$R(FX
)
 <<== |rR(FX
)
|
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\Box$}}}
\gdef\lgwella{\relax}
\gdef\lgwellb{\relax}
\gdef\lgwellc{\relax}
\gdef\lgwelld{\relax}
\gdef\lgwelle{\relax}
\gdef\lgwellf{\relax}
\gdef\lgwellg{\relax}
\gdef\lgwellh{\relax}
\gdef\lgwelli{\relax}
\gdef\lgwellj{\relax}
\gdef\lgwellk{\relax}
\gdef\lgwelll{\relax}
\gdef\lgwellm{\relax}
\gdef\lgwelln{\relax}
\gdef\lgwello{\relax}
\gdef\lgwellp{\relax}
\gdef\lgwellq{\relax}
\gdef\lgwellr{\relax}
\gdef\lgwells{\relax}
\gdef\lgwellt{\relax}
\gdef\lgwellu{\relax}
\gdef\lgwellv{\relax}
\gdef\lgwellw{\relax}
\gdef\lgwellx{\relax}
\gdef\lgwelly{\relax}
\gdef\lgwellz{\relax}
\gdef\lgwellbiga{\relax}
\gdef\lgwellbigb{\relax}
\gdef\lgwellbigc{\relax}
\gdef\lgwellbigd{\relax}
\gdef\lgwellbige{\relax}
\gdef\lgwellbigf{\relax}
\gdef\lgwellbigg{\relax}
\gdef\lgwellbigh{\relax}
\gdef\lgwellbigi{\relax}
\gdef\lgwellbigj{\relax}
\gdef\lgwellbigk{\relax}
\gdef\lgwellbigl{\relax}
\gdef\lgwellbigm{\relax}
\gdef\lgwellbign{\relax}
\gdef\lgwellbigo{\relax}
\gdef\lgwellbigp{\relax}
\gdef\lgwellbigq{\relax}
\gdef\lgwellbigr{\relax}
\gdef\lgwellbigs{\relax}
\gdef\lgwellbigt{\relax}
\gdef\lgwellbigu{\relax}
\gdef\lgwellbigv{\relax}
\gdef\lgwellbigw{\relax}
\gdef\lgwellbigx{\relax}
\gdef\lgwellbigy{\relax}
\gdef\lgwellbigz{\relax}
\endgroup $


$
\mathsf {a}$

venter----


\appendix


\begin {flushleft}$
\mathbf{Priority\ table} 
\newline \mathbf {Preassociative} \newline 
[sup2
] , \linebreak [0] 
[
\mbox{base}
] , \linebreak [0] 
[
\mbox{bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\mbox{big bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\ \$
\ast \$\linebreak[0]\ 
] , \linebreak [0] 
[
\mathbf{flush\ left\ }[ 
\ast 
]
] , \linebreak [0] 
[
\mathsf{x}
] , \linebreak [0] 
[
\mathsf{y}
] , \linebreak [0] 
[
\mathsf{z}
] , \linebreak [0] 
[
[
\ast 
\bowtie
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel{
\ast 
}{\rightarrow}
\ast 
]
] , \linebreak [0] 
[
\mathrm{pyk}
] , \linebreak [0] 
[
\mathrm{tex}
] , \linebreak [0] 
[
\mathrm{name}
] , \linebreak [0] 
[
\mathrm{prio}
] , \linebreak [0] 
[
\ast 
] , \linebreak [0] 
[
\mathsf {T}
] , \linebreak [0] 
[
\mathrm {if}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {
\ast 
}{\Rightarrow }
\ast 
]
] , \linebreak [0] 
[
\mathrm {val}
] , \linebreak [0] 
[
\mathrm {claim}
] , \linebreak [0] 
[
\bot 
] , \linebreak [0] 
[
f(
\ast 
)
] , \linebreak [0] 
[
(
\ast 
){}^{I} 
] , \linebreak [0] 
[
\mathsf {F}
] , \linebreak [0] 
[
\underline {0}
] , \linebreak [0] 
[
\underline {1}
] , \linebreak [0] 
[
\underline {2}
] , \linebreak [0] 
[
\underline {3}
] , \linebreak [0] 
[
\underline {4}
] , \linebreak [0] 
[
\underline {5}
] , \linebreak [0] 
[
\underline {6}
] , \linebreak [0] 
[
\underline {7}
] , \linebreak [0] 
[
\underline {8}
] , \linebreak [0] 
[
\underline {9}
] , \linebreak [0] 
[
0
] , \linebreak [0] 
[
1
] , \linebreak [0] 
[
2
] , \linebreak [0] 
[
3
] , \linebreak [0] 
[
4
] , \linebreak [0] 
[
5
] , \linebreak [0] 
[
6
] , \linebreak [0] 
[
7
] , \linebreak [0] 
[
8
] , \linebreak [0] 
[
9
] , \linebreak [0] 
[
\mathsf {a}
] , \linebreak [0] 
[
\mathsf {b}
] , \linebreak [0] 
[
\mathsf {c}
] , \linebreak [0] 
[
\mathsf {d}
] , \linebreak [0] 
[
\mathsf {e}
] , \linebreak [0] 
[
\mathsf {f}
] , \linebreak [0] 
[
\mathsf {g}
] , \linebreak [0] 
[
\mathsf {h}
] , \linebreak [0] 
[
\mathsf {i}
] , \linebreak [0] 
[
\mathsf {j}
] , \linebreak [0] 
[
\mathsf {k}
] , \linebreak [0] 
[
\mathsf {l}
] , \linebreak [0] 
[
\mathsf {m}
] , \linebreak [0] 
[
\mathsf {n}
] , \linebreak [0] 
[
\mathsf {o}
] , \linebreak [0] 
[
\mathsf {p}
] , \linebreak [0] 
[
\mathsf {q}
] , \linebreak [0] 
[
\mathsf {r}
] , \linebreak [0] 
[
\mathsf {s}
] , \linebreak [0] 
[
\mathsf {t}
] , \linebreak [0] 
[
\mathsf {u}
] , \linebreak [0] 
[
\mathsf {v}
] , \linebreak [0] 
[
\mathsf {w}
] , \linebreak [0] 
[
(
\ast 
)^M
] , \linebreak [0] 
[
\mathrm {If}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[\mathrm {array}\{
\ast 
\}
\ast 
\mathrm {end\ array}
] , \linebreak [0] 
[
l
] , \linebreak [0] 
[
c
] , \linebreak [0] 
[
r
] , \linebreak [0] 
[
\mathrm {empty}
] , \linebreak [0] 
[
\langle 
\ast 
\,{|}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
{\cal M}(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal U}}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}^M(
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}_1(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
identifier( 
\ast 
)
] , \linebreak [0] 
[
identifier_{1}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]remove( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]put( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]add( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rack
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {vector}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {bibliography}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {dictionary}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {body}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {codex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {expansion}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {code}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {cache}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {diagnose}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pyk}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {tex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {texname}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {value}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {message}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {macro}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {definition}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {unpack}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {claim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {priority}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {lambda}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {apply}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {true}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {if}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {quote}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {proclaim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {define}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {introduce}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {hide}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pre}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {post}\mbox {\tt \char34}
] , \linebreak [0] 
[
{\cal E}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {lookup}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {abstract}(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\lceil 
\ast 
\rceil 
] , \linebreak [0] 
[
{\cal M}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathrm {macro}
] , \linebreak [0] 
[
s_0
] , \linebreak [0] 
[
\mathbf {zip}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {assoc}_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[(
\ast 
)^{\mathbf {p}}
] , \linebreak [0] 
[
\mbox{self}
] , \linebreak [0] 
[
[
\ast 
\mathrel {\ddot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\dot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\acute {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {pyk}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {tex}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {name}}{=} 
\ast 
]
] , \linebreak [0] 
[
\mathbf{Priority\ table} [ 
\ast 
]
] , \linebreak [0] 
[
\tilde {{\cal M}}_1
] , \linebreak [0] 
[
\tilde {{\cal M}}_2(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_3(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
(
\ast 
)
] , \linebreak [0] 
[
\left( 
\ast 
\right) 
] , \linebreak [0] 
[
display(
\ast 
)
] , \linebreak [0] 
[
statement(
\ast 
)
] , \linebreak [0] 
[
[
\ast 
]^{\cdot}
] , \linebreak [0] 
[
[
\ast 
]^-
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\langle 
\ast 
\rangle 
] , \linebreak [0] 
[
\mathbf {tuple}_1( 
\ast 
) 
] , \linebreak [0] 
[
\mathbf {tuple}_2( 
\ast 
) 
] , \linebreak [0] 
[
let_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
let_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {claim}{=}
\ast 
]
] , \linebreak [0] 
[
\mbox{checker}
] , \linebreak [0] 
[
\mathbf{check}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_3( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\cdot} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{-} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\circ} 
] , \linebreak [0] 
[
msg
] , \linebreak [0] 
[
[
\ast 
\stackrel {msg}{=} 
\ast 
]
] , \linebreak [0] 
[
{<}stmt{>}
] , \linebreak [0] 
[
stmt
] , \linebreak [0] 
[
[
\ast 
\stackrel {stmt}{=}
\ast 
]
] , \linebreak [0] 
[
HeadNil'
] , \linebreak [0] 
[
HeadPair'
] , \linebreak [0] 
[
Transitivity'
] , \linebreak [0] 
[
{\makebox [0mm][l]{$\bot $}\,{\bot }}
] , \linebreak [0] 
[
Contra'
] , \linebreak [0] 
[
T'_{E}
] , \linebreak [0] 
[
L_{1}
] , \linebreak [0] 
[\underline{
\ast }
] , \linebreak [0] 
[{\cal A}
] , \linebreak [0] 
[{\cal B}
] , \linebreak [0] 
[{\cal C}
] , \linebreak [0] 
[{\cal D}
] , \linebreak [0] 
[{\cal E}
] , \linebreak [0] 
[{\cal F}
] , \linebreak [0] 
[{\cal G}
] , \linebreak [0] 
[{\cal H}
] , \linebreak [0] 
[{\cal I}
] , \linebreak [0] 
[{\cal J}
] , \linebreak [0] 
[{\cal K}
] , \linebreak [0] 
[{\cal L}
] , \linebreak [0] 
[{\cal M}
] , \linebreak [0] 
[{\cal N}
] , \linebreak [0] 
[{\cal O}
] , \linebreak [0] 
[{\cal P}
] , \linebreak [0] 
[{\cal Q}
] , \linebreak [0] 
[{\cal R}
] , \linebreak [0] 
[{\cal S}
] , \linebreak [0] 
[{\cal T}
] , \linebreak [0] 
[{\cal U}
] , \linebreak [0] 
[{\cal V}
] , \linebreak [0] 
[{\cal W}
] , \linebreak [0] 
[{\cal X}
] , \linebreak [0] 
[{\cal Y}
] , \linebreak [0] 
[{\cal Z}
] , \linebreak [0] 
[
\langle 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\langle ^ { \ast } 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\emptyset 
] , \linebreak [0] 
[
Remainder
] , \linebreak [0] 
[
( 
\ast 
)^{\bf v}
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error_{2}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
proof( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
proof_{2}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{I}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\rhd}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\rhd}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{E}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{E}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{+}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{+}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{-}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{-}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\ast}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\char64}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\char64}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\vdash}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\vdash}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{i.e.}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\forall}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\forall}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{;}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{;}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{;}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal T}( 
\ast 
)
] , \linebreak [0] 
[
claims( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
claims_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{<}proof{>}
] , \linebreak [0] 
[
proof
] , \linebreak [0] 
[
[ \mathbf{Lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ \mathbf{Proof\ of\ }
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antilemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ rule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antirule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
\mbox{verifier}
] , \linebreak [0] 
[
{\cal V}_1( 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_3( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_4( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_5( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_6( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_7( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Cut( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Head_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
Tail_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
rule_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rule( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Rule\ tactic
] , \linebreak [0] 
[
Plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[ \mathbf{Theory\ } 
\ast 
]
] , \linebreak [0] 
[
theory_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_3( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_4( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
HeadNil''
] , \linebreak [0] 
[
HeadPair''
] , \linebreak [0] 
[
Transitivity''
] , \linebreak [0] 
[
Contra''
] , \linebreak [0] 
[
HeadNil
] , \linebreak [0] 
[
HeadPair
] , \linebreak [0] 
[
Transitivity
] , \linebreak [0] 
[
Contra
] , \linebreak [0] 
[
T_E
] , \linebreak [0] 
[
ragged\ right
] , \linebreak [0] 
[
ragged\ right\ expansion\ 
] , \linebreak [0] 
[
parm(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
parm^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify^*(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify_2(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[L_a
] , \linebreak [0] 
[L_b
] , \linebreak [0] 
[L_c
] , \linebreak [0] 
[L_d
] , \linebreak [0] 
[L_e
] , \linebreak [0] 
[L_f
] , \linebreak [0] 
[L_g
] , \linebreak [0] 
[L_h
] , \linebreak [0] 
[L_i
] , \linebreak [0] 
[L_j
] , \linebreak [0] 
[L_k
] , \linebreak [0] 
[L_l
] , \linebreak [0] 
[L_m
] , \linebreak [0] 
[L_n
] , \linebreak [0] 
[L_o
] , \linebreak [0] 
[L_p
] , \linebreak [0] 
[L_q
] , \linebreak [0] 
[L_r
] , \linebreak [0] 
[L_s
] , \linebreak [0] 
[L_t
] , \linebreak [0] 
[L_u
] , \linebreak [0] 
[L_v
] , \linebreak [0] 
[L_w
] , \linebreak [0] 
[L_x
] , \linebreak [0] 
[L_y
] , \linebreak [0] 
[L_z
] , \linebreak [0] 
[L_A
] , \linebreak [0] 
[L_B
] , \linebreak [0] 
[L_C
] , \linebreak [0] 
[L_D
] , \linebreak [0] 
[L_E
] , \linebreak [0] 
[L_F
] , \linebreak [0] 
[L_G
] , \linebreak [0] 
[L_H
] , \linebreak [0] 
[L_I
] , \linebreak [0] 
[L_J
] , \linebreak [0] 
[L_K
] , \linebreak [0] 
[L_L
] , \linebreak [0] 
[L_M
] , \linebreak [0] 
[L_N
] , \linebreak [0] 
[L_O
] , \linebreak [0] 
[L_P
] , \linebreak [0] 
[L_Q
] , \linebreak [0] 
[L_R
] , \linebreak [0] 
[L_S
] , \linebreak [0] 
[L_T
] , \linebreak [0] 
[L_U
] , \linebreak [0] 
[L_V
] , \linebreak [0] 
[L_W
] , \linebreak [0] 
[L_X
] , \linebreak [0] 
[L_Y
] , \linebreak [0] 
[L_Z
] , \linebreak [0] 
[L_?
] , \linebreak [0] 
[
Reflexivity
] , \linebreak [0] 
[
Reflexivity_1
] , \linebreak [0] 
[
Commutativity
] , \linebreak [0] 
[
Commutativity_1
] , \linebreak [0] 
[
{<}tactic{>}
] , \linebreak [0] 
[
tactic
] , \linebreak [0] 
[
[
\ast 
\stackrel {tactic}{=}
\ast 
]
] , \linebreak [0] 
[
{\cal P}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal P}^*( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
p_0
] , \linebreak [0] 
[
conclude_1 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_2 ( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_3 ( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_4 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mbox{check}
] , \linebreak [0] 
[
[
\ast 
\stackrel {\circ}{=}
\ast 
]
] , \linebreak [0] 
[
RootVisible(
\ast 
)
] , \linebreak [0] 
[
A
] , \linebreak [0] 
[
R
] , \linebreak [0] 
[
C
] , \linebreak [0] 
[
T
] , \linebreak [0] 
[
L
] , \linebreak [0] 
[
\{
\ast 
\}
] , \linebreak [0] 
[\overline{
\ast }
] , \linebreak [0] 
[
\mathit{a}
] , \linebreak [0] 
[
\mathit{b}
] , \linebreak [0] 
[
\mathit{c}
] , \linebreak [0] 
[
\mathit{d}
] , \linebreak [0] 
[
\mathit{e}
] , \linebreak [0] 
[
\mathit{f}
] , \linebreak [0] 
[
\mathit{g}
] , \linebreak [0] 
[
\mathit{h}
] , \linebreak [0] 
[
\mathit{i}
] , \linebreak [0] 
[
\mathit{j}
] , \linebreak [0] 
[
\mathit{k}
] , \linebreak [0] 
[
\mathit{l}
] , \linebreak [0] 
[
\mathit{m}
] , \linebreak [0] 
[
\mathit{n}
] , \linebreak [0] 
[
\mathit{o}
] , \linebreak [0] 
[
\mathit{p}
] , \linebreak [0] 
[
\mathit{q}
] , \linebreak [0] 
[
\mathit{r}
] , \linebreak [0] 
[
\mathit{s}
] , \linebreak [0] 
[
\mathit{t}
] , \linebreak [0] 
[
\mathit{u}
] , \linebreak [0] 
[
\mathit{v}
] , \linebreak [0] 
[
\mathit{w}
] , \linebreak [0] 
[
\mathit{x}
] , \linebreak [0] 
[
\mathit{y}
] , \linebreak [0] 
[
\mathit{z}
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
Ded(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_0(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_7(
\ast 
)
] , \linebreak [0] 
[
Ded_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
S
] , \linebreak [0] 
[
Neg
] , \linebreak [0] 
[
MP
] , \linebreak [0] 
[
Gen
] , \linebreak [0] 
[
Ded
] , \linebreak [0] 
[
S1
] , \linebreak [0] 
[
S2
] , \linebreak [0] 
[
S3
] , \linebreak [0] 
[
S4
] , \linebreak [0] 
[
S5
] , \linebreak [0] 
[
S6
] , \linebreak [0] 
[
S7
] , \linebreak [0] 
[
S8
] , \linebreak [0] 
[
S9
] , \linebreak [0] 
[
Repetition
] , \linebreak [0] 
[
A1'
] , \linebreak [0] 
[
A2'
] , \linebreak [0] 
[
A4'
] , \linebreak [0] 
[
A5'
] , \linebreak [0] 
[
Prop\ 3.2a
] , \linebreak [0] 
[
Prop\ 3.2b
] , \linebreak [0] 
[
Prop\ 3.2c
] , \linebreak [0] 
[
Prop\ 3.2d
] , \linebreak [0] 
[
Prop\ 3.2e_1
] , \linebreak [0] 
[
Prop\ 3.2e_2
] , \linebreak [0] 
[
Prop\ 3.2e
] , \linebreak [0] 
[
Prop\ 3.2f_1
] , \linebreak [0] 
[
Prop\ 3.2f_2
] , \linebreak [0] 
[
Prop\ 3.2f
] , \linebreak [0] 
[
Prop\ 3.2g_1
] , \linebreak [0] 
[
Prop\ 3.2g_2
] , \linebreak [0] 
[
Prop\ 3.2g
] , \linebreak [0] 
[
Prop\ 3.2h_1
] , \linebreak [0] 
[
Prop\ 3.2h_2
] , \linebreak [0] 
[
Prop\ 3.2h
] , \linebreak [0] 
[
Block_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Block_2(
\ast 
)
] , \linebreak [0] 
[kvanti
] , \linebreak [0] 
[UniqueMember
] , \linebreak [0] 
[UniqueMember(Type)
] , \linebreak [0] 
[SameSeries
] , \linebreak [0] 
[A4
] , \linebreak [0] 
[SameMember
] , \linebreak [0] 
[Qclosed(Addition)
] , \linebreak [0] 
[Qclosed(Multiplication)
] , \linebreak [0] 
[FromCartProd(1)
] , \linebreak [0] 
[
\mbox{1rule fromCartProd(2)}
] , \linebreak [0] 
[constantRationalSeries(
\ast 
)
] , \linebreak [0] 
[cartProd(
\ast 
)
] , \linebreak [0] 
[Power(
\ast 
)
] , \linebreak [0] 
[binaryUnion(
\ast 
,
\ast 
)
] , \linebreak [0] 
[SetOfRationalSeries
] , \linebreak [0] 
[IsSubset(
\ast 
,
\ast 
)
] , \linebreak [0] 
[(p
\ast 
,
\ast 
)
] , \linebreak [0] 
[(s
\ast 
)
] , \linebreak [0] 
[(\cdots{})
] , \linebreak [0] 
[\texttt{Objekt-var}
] , \linebreak [0] 
[\texttt{Ex-var}
] , \linebreak [0] 
[\texttt{Ph-var}
] , \linebreak [0] 
[\texttt{V\ae{}rdi}
] , \linebreak [0] 
[\texttt{Variabel}
] , \linebreak [0] 
[Op(
\ast 
)
] , \linebreak [0] 
[Op(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
 \mathrel {\ddot {==}} 
\ast 
] , \linebreak [0] 
[ContainsEmpty(
\ast 
)
] , \linebreak [0] 
[Nat(
\ast 
)
] , \linebreak [0] 
[
Dedu(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_0(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Dedu_{s}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_7(
\ast 
)
] , \linebreak [0] 
[
Dedu_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Ex_{1}
] , \linebreak [0] 
[Ex_{2}
] , \linebreak [0] 
[Ex3
] , \linebreak [0] 
[Ex_{10}
] , \linebreak [0] 
[Ex_{20}
] , \linebreak [0] 
[
\ast 
 _{Ex}
] , \linebreak [0] 
[
\ast 
 ^{Ex}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[ph_{1}
] , \linebreak [0] 
[ph_{2}
] , \linebreak [0] 
[ph_{3}
] , \linebreak [0] 
[
\ast 
 _{Ph} 
] , \linebreak [0] 
[
\ast 
 ^{Ph}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me} 
] , \linebreak [0] 
[\mathsf {bs}
] , \linebreak [0] 
[ \mathsf {OBS}
] , \linebreak [0] 
[{\cal BS}
] , \linebreak [0] 
[\mathrm{\O}
] , \linebreak [0] 
[SystemQ
] , \linebreak [0] 
[MP
] , \linebreak [0] 
[Gen
] , \linebreak [0] 
[Repetition
] , \linebreak [0] 
[Neg
] , \linebreak [0] 
[Ded
] , \linebreak [0] 
[ExistIntro
] , \linebreak [0] 
[Extensionality
] , \linebreak [0] 
[\O{}def
] , \linebreak [0] 
[PairDef
] , \linebreak [0] 
[UnionDef
] , \linebreak [0] 
[PowerDef
] , \linebreak [0] 
[SeparationDef
] , \linebreak [0] 
[AddDoubleNeg
] , \linebreak [0] 
[RemoveDoubleNeg
] , \linebreak [0] 
[AndCommutativity
] , \linebreak [0] 
[AutoImply
] , \linebreak [0] 
[Contrapositive
] , \linebreak [0] 
[FirstConjunct
] , \linebreak [0] 
[SecondConjunct
] , \linebreak [0] 
[FromContradiction
] , \linebreak [0] 
[FromDisjuncts
] , \linebreak [0] 
[IffCommutativity
] , \linebreak [0] 
[IffFirst
] , \linebreak [0] 
[IffSecond
] , \linebreak [0] 
[ImplyTransitivity
] , \linebreak [0] 
[JoinConjuncts
] , \linebreak [0] 
[MP2
] , \linebreak [0] 
[MP3
] , \linebreak [0] 
[MP4
] , \linebreak [0] 
[MP5
] , \linebreak [0] 
[MT
] , \linebreak [0] 
[NegativeMT
] , \linebreak [0] 
[Technicality
] , \linebreak [0] 
[Weakening
] , \linebreak [0] 
[WeakenOr1
] , \linebreak [0] 
[WeakenOr2
] , \linebreak [0] 
[Formula2Pair
] , \linebreak [0] 
[Pair2Formula
] , \linebreak [0] 
[Formula2Union
] , \linebreak [0] 
[Union2Formula
] , \linebreak [0] 
[Formula2Sep
] , \linebreak [0] 
[Sep2Formula
] , \linebreak [0] 
[Formula2Power
] , \linebreak [0] 
[SubsetInPower
] , \linebreak [0] 
[HelperPowerIsSub
] , \linebreak [0] 
[PowerIsSub
] , \linebreak [0] 
[(Switch)HelperPowerIsSub
] , \linebreak [0] 
[(Switch)PowerIsSub
] , \linebreak [0] 
[ToSetEquality
] , \linebreak [0] 
[HelperToSetEquality(t)
] , \linebreak [0] 
[ToSetEquality(t)
] , \linebreak [0] 
[HelperFromSetEquality
] , \linebreak [0] 
[FromSetEquality
] , \linebreak [0] 
[HelperReflexivity
] , \linebreak [0] 
[Reflexivity
] , \linebreak [0] 
[HelperSymmetry
] , \linebreak [0] 
[Symmetry
] , \linebreak [0] 
[HelperTransitivity
] , \linebreak [0] 
[Transitivity
] , \linebreak [0] 
[ERisReflexive
] , \linebreak [0] 
[ERisSymmetric
] , \linebreak [0] 
[ERisTransitive
] , \linebreak [0] 
[\O{}isSubset
] , \linebreak [0] 
[HelperMemberNot\O{}
] , \linebreak [0] 
[MemberNot\O{}
] , \linebreak [0] 
[HelperUnique\O{}
] , \linebreak [0] 
[Unique\O{}
] , \linebreak [0] 
[==\!{}Reflexivity
] , \linebreak [0] 
[==\!{}Symmetry
] , \linebreak [0] 
[Helper\!{}==\!{}Transitivity
] , \linebreak [0] 
[\!{}==\!{}Transitivity
] , \linebreak [0] 
[HelperTransferNotEq
] , \linebreak [0] 
[TransferNotEq
] , \linebreak [0] 
[HelperPairSubset
] , \linebreak [0] 
[Helper(2)PairSubset
] , \linebreak [0] 
[PairSubset
] , \linebreak [0] 
[SamePair
] , \linebreak [0] 
[SameSingleton
] , \linebreak [0] 
[UnionSubset
] , \linebreak [0] 
[SameUnion
] , \linebreak [0] 
[SeparationSubset
] , \linebreak [0] 
[SameSeparation
] , \linebreak [0] 
[SameBinaryUnion
] , \linebreak [0] 
[IntersectionSubset
] , \linebreak [0] 
[SameIntersection
] , \linebreak [0] 
[AutoMember
] , \linebreak [0] 
[HelperEqSysNot\O{}
] , \linebreak [0] 
[EqSysNot\O{}
] , \linebreak [0] 
[HelperEqSubset
] , \linebreak [0] 
[EqSubset
] , \linebreak [0] 
[HelperEqNecessary
] , \linebreak [0] 
[EqNecessary
] , \linebreak [0] 
[HelperNoneEqNecessary
] , \linebreak [0] 
[Helper(2)NoneEqNecessary
] , \linebreak [0] 
[NoneEqNecessary
] , \linebreak [0] 
[EqClassIsSubset
] , \linebreak [0] 
[EqClassesAreDisjoint
] , \linebreak [0] 
[AllDisjoint
] , \linebreak [0] 
[AllDisjointImply
] , \linebreak [0] 
[BSsubset
] , \linebreak [0] 
[Union(BS/R)subset
] , \linebreak [0] 
[UnionIdentity
] , \linebreak [0] 
[EqSysIsPartition
] , \linebreak [0] 
[(x1)
] , \linebreak [0] 
[(x2)
] , \linebreak [0] 
[(y1)
] , \linebreak [0] 
[(y2)
] , \linebreak [0] 
[(v1)
] , \linebreak [0] 
[(v2)
] , \linebreak [0] 
[(v3)
] , \linebreak [0] 
[(v4)
] , \linebreak [0] 
[(v2n)
] , \linebreak [0] 
[(m1)
] , \linebreak [0] 
[(m2)
] , \linebreak [0] 
[(n1)
] , \linebreak [0] 
[(n2)
] , \linebreak [0] 
[(n3)
] , \linebreak [0] 
[(\epsilon)
] , \linebreak [0] 
[(\epsilon)_{1}
] , \linebreak [0] 
[(\epsilon 2)
] , \linebreak [0] 
[(fep)
] , \linebreak [0] 
[(fx)
] , \linebreak [0] 
[(fy)
] , \linebreak [0] 
[(fz)
] , \linebreak [0] 
[(fu)
] , \linebreak [0] 
[(fv)
] , \linebreak [0] 
[(fw)
] , \linebreak [0] 
[(rx)
] , \linebreak [0] 
[(ry)
] , \linebreak [0] 
[(rz)
] , \linebreak [0] 
[(ru)
] , \linebreak [0] 
[(sx)
] , \linebreak [0] 
[(sx1)
] , \linebreak [0] 
[(sy)
] , \linebreak [0] 
[(sy1)
] , \linebreak [0] 
[(sz)
] , \linebreak [0] 
[(sz1)
] , \linebreak [0] 
[(su)
] , \linebreak [0] 
[(su1)
] , \linebreak [0] 
[(fxs)
] , \linebreak [0] 
[(fys)
] , \linebreak [0] 
[(crs1)
] , \linebreak [0] 
[(f1)
] , \linebreak [0] 
[(f2)
] , \linebreak [0] 
[(f3)
] , \linebreak [0] 
[(f4)
] , \linebreak [0] 
[(op1)
] , \linebreak [0] 
[(op2)
] , \linebreak [0] 
[(r1)
] , \linebreak [0] 
[(s1)
] , \linebreak [0] 
[(s2)
] , \linebreak [0] 
[X_{1}
] , \linebreak [0] 
[X_{2}
] , \linebreak [0] 
[Y_{1}
] , \linebreak [0] 
[Y_{2}
] , \linebreak [0] 
[V_{1}
] , \linebreak [0] 
[V_{2}
] , \linebreak [0] 
[V_{3}
] , \linebreak [0] 
[V_{4}
] , \linebreak [0] 
[V_{2n}
] , \linebreak [0] 
[M_{1}
] , \linebreak [0] 
[M_{2}
] , \linebreak [0] 
[N_{1} 
] , \linebreak [0] 
[N_{2} 
] , \linebreak [0] 
[N_{3} 
] , \linebreak [0] 
[\epsilon
] , \linebreak [0] 
[\epsilon 1
] , \linebreak [0] 
[\epsilon 2
] , \linebreak [0] 
[FX
] , \linebreak [0] 
[FY
] , \linebreak [0] 
[FZ
] , \linebreak [0] 
[FU
] , \linebreak [0] 
[FV
] , \linebreak [0] 
[FW
] , \linebreak [0] 
[FEP
] , \linebreak [0] 
[RX
] , \linebreak [0] 
[RY
] , \linebreak [0] 
[RZ
] , \linebreak [0] 
[RU
] , \linebreak [0] 
[(SX)
] , \linebreak [0] 
[(SX1)
] , \linebreak [0] 
[(SY)
] , \linebreak [0] 
[(SY1)
] , \linebreak [0] 
[(SZ)
] , \linebreak [0] 
[(SZ1)
] , \linebreak [0] 
[(SU)
] , \linebreak [0] 
[(SU1)
] , \linebreak [0] 
[FXS
] , \linebreak [0] 
[FYS
] , \linebreak [0] 
[(F1)
] , \linebreak [0] 
[(F2)
] , \linebreak [0] 
[(F3)
] , \linebreak [0] 
[(F4)
] , \linebreak [0] 
[(OP1)
] , \linebreak [0] 
[(OP2)
] , \linebreak [0] 
[(R1)
] , \linebreak [0] 
[(S1)
] , \linebreak [0] 
[(S2)
] , \linebreak [0] 
[(EPob)
] , \linebreak [0] 
[(CRS1ob)
] , \linebreak [0] 
[(F1ob)
] , \linebreak [0] 
[(F2ob)
] , \linebreak [0] 
[(F3ob)
] , \linebreak [0] 
[(F4ob)
] , \linebreak [0] 
[(N1ob)
] , \linebreak [0] 
[(N2ob)
] , \linebreak [0] 
[(OP1ob)
] , \linebreak [0] 
[(OP2ob)
] , \linebreak [0] 
[(R1ob)
] , \linebreak [0] 
[(S1ob)
] , \linebreak [0] 
[(S2ob)
] , \linebreak [0] 
[ph_{4}
] , \linebreak [0] 
[ph_{5}
] , \linebreak [0] 
[ph_{6}
] , \linebreak [0] 
[NAT
] , \linebreak [0] 
[RATIONAL_SERIES
] , \linebreak [0] 
[SERIES
] , \linebreak [0] 
[SetOfReals
] , \linebreak [0] 
[SetOfFxs
] , \linebreak [0] 
[N
] , \linebreak [0] 
[Q
] , \linebreak [0] 
[X
] , \linebreak [0] 
[xs
] , \linebreak [0] 
[xaF
] , \linebreak [0] 
[ysF
] , \linebreak [0] 
[us
] , \linebreak [0] 
[usFoelge
] , \linebreak [0] 
[0
] , \linebreak [0] 
[1
] , \linebreak [0] 
[(-1)
] , \linebreak [0] 
[2
] , \linebreak [0] 
[3
] , \linebreak [0] 
[1/2
] , \linebreak [0] 
[1/3
] , \linebreak [0] 
[2/3
] , \linebreak [0] 
[0f
] , \linebreak [0] 
[
\mbox{1f}
] , \linebreak [0] 
[00
] , \linebreak [0] 
[
\mbox{01}
] , \linebreak [0] 
[(--01)
] , \linebreak [0] 
[02
] , \linebreak [0] 
[01//02
] , \linebreak [0] 
[PlusAssociativity(R)
] , \linebreak [0] 
[PlusAssociativity(R)XX
] , \linebreak [0] 
[Plus0(R)
] , \linebreak [0] 
[Negative(R)
] , \linebreak [0] 
[Times1(R)
] , \linebreak [0] 
[lessAddition(R)
] , \linebreak [0] 
[PlusCommutativity(R)
] , \linebreak [0] 
[LeqAntisymmetry(R)
] , \linebreak [0] 
[LeqTransitivity(R)
] , \linebreak [0] 
[leqAddition(R)
] , \linebreak [0] 
[Distribution(R)
] , \linebreak [0] 
[A4(Axiom)
] , \linebreak [0] 
[InductionAxiom
] , \linebreak [0] 
[EqualityAxiom
] , \linebreak [0] 
[EqLeqAxiom
] , \linebreak [0] 
[EqAdditionAxiom
] , \linebreak [0] 
[EqMultiplicationAxiom
] , \linebreak [0] 
[QisClosed(Reciprocal)(Imply)
] , \linebreak [0] 
[QisClosed(Reciprocal)
] , \linebreak [0] 
[QisClosed(Negative)(Imply)
] , \linebreak [0] 
[QisClosed(Negative)
] , \linebreak [0] 
[leqReflexivity
] , \linebreak [0] 
[leqAntisymmetryAxiom
] , \linebreak [0] 
[leqTransitivityAxiom
] , \linebreak [0] 
[leqTotality
] , \linebreak [0] 
[leqAdditionAxiom
] , \linebreak [0] 
[leqMultiplicationAxiom
] , \linebreak [0] 
[plusAssociativity
] , \linebreak [0] 
[plusCommutativity
] , \linebreak [0] 
[Negative
] , \linebreak [0] 
[plus0
] , \linebreak [0] 
[timesAssociativity
] , \linebreak [0] 
[timesCommutativity
] , \linebreak [0] 
[ReciprocalAxiom
] , \linebreak [0] 
[times1
] , \linebreak [0] 
[Distribution
] , \linebreak [0] 
[0not1
] , \linebreak [0] 
[
\mbox{lemma eqLeq(R)}
] , \linebreak [0] 
[TimesAssociativity(R)
] , \linebreak [0] 
[TimesCommutativity(R)
] , \linebreak [0] 
[(Adgic)SameR
] , \linebreak [0] 
[Separation2formula(1)
] , \linebreak [0] 
[Separation2formula(2)
] , \linebreak [0] 
[Cauchy
] , \linebreak [0] 
[PlusF
] , \linebreak [0] 
[ReciprocalF
] , \linebreak [0] 
[From==
] , \linebreak [0] 
[To==
] , \linebreak [0] 
[FromInR
] , \linebreak [0] 
[PlusR(Sym)
] , \linebreak [0] 
[ReciprocalR(Axiom)
] , \linebreak [0] 
[LessMinus1(N)
] , \linebreak [0] 
[Nonnegative(N)
] , \linebreak [0] 
[US0
] , \linebreak [0] 
[NextXS(UpperBound)
] , \linebreak [0] 
[NextXS(NoUpperBound)
] , \linebreak [0] 
[NextUS(UpperBound)
] , \linebreak [0] 
[NextUS(NoUpperBound)
] , \linebreak [0] 
[ExpZero
] , \linebreak [0] 
[ExpPositive
] , \linebreak [0] 
[ExpZero(R)
] , \linebreak [0] 
[ExpPositive(R)
] , \linebreak [0] 
[BSzero
] , \linebreak [0] 
[BSpositive
] , \linebreak [0] 
[UStelescope(Zero)
] , \linebreak [0] 
[UStelescope(Positive)
] , \linebreak [0] 
[EqAddition(R)
] , \linebreak [0] 
[FromLimit
] , \linebreak [0] 
[ToUpperBound
] , \linebreak [0] 
[FromUpperBound
] , \linebreak [0] 
[USisUpperBound
] , \linebreak [0] 
[0not1(R)
] , \linebreak [0] 
[ExpUnbounded(R)
] , \linebreak [0] 
[FromLeq(Advanced)(N)
] , \linebreak [0] 
[FromLeastUpperBound
] , \linebreak [0] 
[ToLeastUpperBound
] , \linebreak [0] 
[XSisNotUpperBound
] , \linebreak [0] 
[ysFGreater
] , \linebreak [0] 
[ysFLess
] , \linebreak [0] 
[SmallInverse
] , \linebreak [0] 
[NatType
] , \linebreak [0] 
[RationalType
] , \linebreak [0] 
[SeriesType
] , \linebreak [0] 
[Max
] , \linebreak [0] 
[Numerical
] , \linebreak [0] 
[NumericalF
] , \linebreak [0] 
[MemberOfSeries(Imply)
] , \linebreak [0] 
[JoinConjuncts(2conditions)
] , \linebreak [0] 
[
\mbox{prop lemma imply negation}
] , \linebreak [0] 
[TND
] , \linebreak [0] 
[FromNegatedImply
] , \linebreak [0] 
[ToNegatedImply
] , \linebreak [0] 
[FromNegated(2*Imply)
] , \linebreak [0] 
[FromNegatedAnd
] , \linebreak [0] 
[FromNegatedOr
] , \linebreak [0] 
[ToNegatedOr
] , \linebreak [0] 
[FromNegations
] , \linebreak [0] 
[From3Disjuncts
] , \linebreak [0] 
[From2*2Disjuncts
] , \linebreak [0] 
[NegateDisjunct1
] , \linebreak [0] 
[NegateDisjunct2
] , \linebreak [0] 
[ExpandDisjuncts
] , \linebreak [0] 
[SENC1
] , \linebreak [0] 
[SENC2
] , \linebreak [0] 
[LessLeq(R)
] , \linebreak [0] 
[MemberOfSeries
] , \linebreak [0] 
[memberOfSeries(Type)
] , \linebreak [0] 
[sup
] , \linebreak [0] 
[ToNegatedAnd(1)
] , \linebreak [0] 
[UniqueNegative
] , \linebreak [0] 
[DoubleMinus
] , \linebreak [0] 
[MinusNegated
] , \linebreak [0] 
[eqReflexivity
] , \linebreak [0] 
[eqSymmetry
] , \linebreak [0] 
[eqTransitivity
] , \linebreak [0] 
[eqTransitivity4
] , \linebreak [0] 
[eqTransitivity5
] , \linebreak [0] 
[eqTransitivity6
] , \linebreak [0] 
[AddEquations
] , \linebreak [0] 
[SubtractEquations
] , \linebreak [0] 
[SubtractEquationsLeft
] , \linebreak [0] 
[MultiplyEquations
] , \linebreak [0] 
[EqNegated
] , \linebreak [0] 
[PositiveToRight(Eq)
] , \linebreak [0] 
[PositiveToLeft(Eq)(1 term)
] , \linebreak [0] 
[NegativeToLeft(Eq)
] , \linebreak [0] 
[NonreciprocalToRight(Eq)(1 term)
] , \linebreak [0] 
[PlusAssociativity(4 terms)
] , \linebreak [0] 
[LessNeq
] , \linebreak [0] 
[NeqSymmetry
] , \linebreak [0] 
[NeqNegated
] , \linebreak [0] 
[SubNeqRight
] , \linebreak [0] 
[SubNeqLeft
] , \linebreak [0] 
[NegativeToRight(Neq)(1 term)
] , \linebreak [0] 
[NeqAddition
] , \linebreak [0] 
[NeqMultiplication
] , \linebreak [0] 
[NonzeroProduct(2)
] , \linebreak [0] 
[UStelescope(+1)
] , \linebreak [0] 
[TelescopeBound(Base)
] , \linebreak [0] 
[TelescopeBound(Indu)
] , \linebreak [0] 
[TelescopeBound
] , \linebreak [0] 
[IntervalSize(Base)
] , \linebreak [0] 
[IntervalSize(Indu)
] , \linebreak [0] 
[IntervalSize
] , \linebreak [0] 
[XS<US
] , \linebreak [0] 
[
\mbox{lemma USdecreasing(+1)}
] , \linebreak [0] 
[CloseUS
] , \linebreak [0] 
[CloseUS(n+1)
] , \linebreak [0] 
[AllNegated(Imply)
] , \linebreak [0] 
[ExistNegated(Imply)
] , \linebreak [0] 
[IntroExist(Helper)
] , \linebreak [0] 
[IntroExist
] , \linebreak [0] 
[ExistMP
] , \linebreak [0] 
[ExistMP2
] , \linebreak [0] 
[TwiceExistMP
] , \linebreak [0] 
[TwiceExistMP2
] , \linebreak [0] 
[EAE-MP
] , \linebreak [0] 
[AddAll 
] , \linebreak [0] 
[AddExist(Helper1)
] , \linebreak [0] 
[AddExist(Helper2)
] , \linebreak [0] 
[AddExist
] , \linebreak [0] 
[AddExist(SimpleAnt)
] , \linebreak [0] 
[AddExist(Simple)
] , \linebreak [0] 
[AddEAE
] , \linebreak [0] 
[AEA-negated
] , \linebreak [0] 
[EEA-negated
] , \linebreak [0] 
[Induction
] , \linebreak [0] 
[leqAntisymmetry
] , \linebreak [0] 
[leqTransitivity
] , \linebreak [0] 
[leqAddition
] , \linebreak [0] 
[leqMultiplication
] , \linebreak [0] 
[Reciprocal
] , \linebreak [0] 
[Equality
] , \linebreak [0] 
[eqLeq
] , \linebreak [0] 
[eqAddition
] , \linebreak [0] 
[eqMultiplication
] , \linebreak [0] 
[LeqMultiplicationLeft 
] , \linebreak [0] 
[LeqLessEq
] , \linebreak [0] 
[LessLeq
] , \linebreak [0] 
[FromLeqGeq
] , \linebreak [0] 
[subLeqRight
] , \linebreak [0] 
[subLeqLeft
] , \linebreak [0] 
[Leq+1
] , \linebreak [0] 
[PositiveToRight(Leq)
] , \linebreak [0] 
[PositiveToRight(Leq)(1 term)
] , \linebreak [0] 
[
\mbox{lemma negativeToRight(Leq)}
] , \linebreak [0] 
[PositiveToLeft(Leq)
] , \linebreak [0] 
[negativeToLeft(Leq)
] , \linebreak [0] 
[negativeToLeft(Leq)(1 term)
] , \linebreak [0] 
[LeqAdditionLeft
] , \linebreak [0] 
[leqSubtraction
] , \linebreak [0] 
[leqSubtractionLeft
] , \linebreak [0] 
[thirdGeq
] , \linebreak [0] 
[LeqNegated
] , \linebreak [0] 
[AddEquations(Leq)
] , \linebreak [0] 
[MultiplyEquations(Leq)
] , \linebreak [0] 
[ThirdGeqSeries
] , \linebreak [0] 
[LeqNeqLess
] , \linebreak [0] 
[FromLess
] , \linebreak [0] 
[ToLess
] , \linebreak [0] 
[fromNotLess
] , \linebreak [0] 
[toNotLess
] , \linebreak [0] 
[NegativeLessPositive
] , \linebreak [0] 
[leqLessTransitivity
] , \linebreak [0] 
[LessLeqTransitivity
] , \linebreak [0] 
[LessTransitivity
] , \linebreak [0] 
[LessTotality
] , \linebreak [0] 
[SubLessRight
] , \linebreak [0] 
[SubLessLeft
] , \linebreak [0] 
[SwitchTerms(x<y-z)
] , \linebreak [0] 
[SwitchTerms(x-y<z)
] , \linebreak [0] 
[LessAddition
] , \linebreak [0] 
[LessAdditionLeft
] , \linebreak [0] 
[LessMultiplication
] , \linebreak [0] 
[LessMultiplicationLeft
] , \linebreak [0] 
[LessDivision
] , \linebreak [0] 
[PositiveToRight(Less)
] , \linebreak [0] 
[PositiveToLeft(Less)
] , \linebreak [0] 
[NegativeToLeft(Less)
] , \linebreak [0] 
[NegativeToRight(Less)
] , \linebreak [0] 
[AddEquations(Less)
] , \linebreak [0] 
[AddEquations(LeqLess)
] , \linebreak [0] 
[reciprocalToLeft(Less)
] , \linebreak [0] 
[LessNegated
] , \linebreak [0] 
[PositiveNonzero
] , \linebreak [0] 
[PositiveNegated
] , \linebreak [0] 
[NonpositiveNegated
] , \linebreak [0] 
[NegativeNegated
] , \linebreak [0] 
[NonnegativeNegated
] , \linebreak [0] 
[PositiveHalved
] , \linebreak [0] 
[PositiveInverted
] , \linebreak [0] 
[NonnegativeNumerical
] , \linebreak [0] 
[NegativeNumerical
] , \linebreak [0] 
[PositiveNumerical
] , \linebreak [0] 
[
\mbox{lemma nonpositiveNumerical}
] , \linebreak [0] 
[|0|=0
] , \linebreak [0] 
[0<=|x|
] , \linebreak [0] 
[x<=|x|
] , \linebreak [0] 
[FromPositiveNumerical
] , \linebreak [0] 
[SameNumerical
] , \linebreak [0] 
[SignNumerical(+)
] , \linebreak [0] 
[SignNumerical
] , \linebreak [0] 
[ToNumericalLess
] , \linebreak [0] 
[FromNumericalGreater
] , \linebreak [0] 
[NumericalDifference
] , \linebreak [0] 
[NumericalDifferenceLess(Helper)
] , \linebreak [0] 
[NumericalDifferenceLess
] , \linebreak [0] 
[SplitNumericalSumHelper
] , \linebreak [0] 
[splitNumericalSum(++)
] , \linebreak [0] 
[splitNumericalSum(--)
] , \linebreak [0] 
[splitNumericalSum(+-small)
] , \linebreak [0] 
[splitNumericalSum(+-big)
] , \linebreak [0] 
[splitNumericalSum(+-)
] , \linebreak [0] 
[splitNumericalSum(-+)
] , \linebreak [0] 
[splitNumericalSum
] , \linebreak [0] 
[SplitNumericalProduct(++)
] , \linebreak [0] 
[SplitNumericalProduct(+-)
] , \linebreak [0] 
[SplitNumericalProduct
] , \linebreak [0] 
[insertMiddleTerm(Numerical)
] , \linebreak [0] 
[insertTwoMiddleTerms(Numerical)
] , \linebreak [0] 
[Three2twoTerms
] , \linebreak [0] 
[Three2threeTerms
] , \linebreak [0] 
[Three2twoFactors
] , \linebreak [0] 
[Three2threeFactors
] , \linebreak [0] 
[Times(-1)
] , \linebreak [0] 
[Times(-1)Left
] , \linebreak [0] 
[MaxLeq(1)
] , \linebreak [0] 
[MaxLeq(2)
] , \linebreak [0] 
[LessThanMax
] , \linebreak [0] 
[x+y=zBackwards
] , \linebreak [0] 
[x*y=zBackwards
] , \linebreak [0] 
[x=x+(y-y)
] , \linebreak [0] 
[x=x+y-y
] , \linebreak [0] 
[x=x*y*(1/y)
] , \linebreak [0] 
[insertMiddleTerm(Sum)
] , \linebreak [0] 
[insertTwoMiddleTerms(Sum)
] , \linebreak [0] 
[insertMiddleTerm(Difference)
] , \linebreak [0] 
[x*0+x=x
] , \linebreak [0] 
[x*0=0
] , \linebreak [0] 
[NonnegativeFactors
] , \linebreak [0] 
[NonzeroFactors
] , \linebreak [0] 
[PositiveFactors
] , \linebreak [0] 
[PlusTimesMinus
] , \linebreak [0] 
[MinusTimesMinus
] , \linebreak [0] 
[(-1)*(-1)+(-1)*1=0
] , \linebreak [0] 
[(-1)*(-1)=1
] , \linebreak [0] 
[0<1Helper
] , \linebreak [0] 
[0<1
] , \linebreak [0] 
[0<2
] , \linebreak [0] 
[0<3
] , \linebreak [0] 
[0<1/2
] , \linebreak [0] 
[0<1/3
] , \linebreak [0] 
[TwoWholes
] , \linebreak [0] 
[ThreeWholes
] , \linebreak [0] 
[TwoHalves
] , \linebreak [0] 
[ThreeThirds
] , \linebreak [0] 
[-x-y=-(x+y)
] , \linebreak [0] 
[-x*y=-(x*y)
] , \linebreak [0] 
[-0=0
] , \linebreak [0] 
[SFsymmetry
] , \linebreak [0] 
[SFtransitivity
] , \linebreak [0] 
[f2R(Plus)
] , \linebreak [0] 
[f2R(Times)
] , \linebreak [0] 
[<<TransitivityHelper(Q)
] , \linebreak [0] 
[<<Transitivity
] , \linebreak [0] 
[<<==Reflexivity
] , \linebreak [0] 
[<<==AntisymmetryHelper(Q)
] , \linebreak [0] 
[FromNot<f(Weak)(Helper)
] , \linebreak [0] 
[FromNot<f(Weak)
] , \linebreak [0] 
[FromNot<f(Strong)(Helper2)
] , \linebreak [0] 
[FromNot<f(Strong)(Helper)
] , \linebreak [0] 
[FromNot<f(Strong)
] , \linebreak [0] 
[fromNotSameF(Strongest)(Helper2)
] , \linebreak [0] 
[fromNotSameF(Strongest)(Helper)
] , \linebreak [0] 
[fromNotSameF(Strongest)
] , \linebreak [0] 
[ToLess(F)(Helper)
] , \linebreak [0] 
[ToLess(F)
] , \linebreak [0] 
[FromNot<<
] , \linebreak [0] 
[ToLess(R)
] , \linebreak [0] 
[FromNotSameF(Weak)(Helper)
] , \linebreak [0] 
[FromNotSameF(Weak)
] , \linebreak [0] 
[FromNotLess(F)
] , \linebreak [0] 
[==Addition
] , \linebreak [0] 
[==AdditionLeft
] , \linebreak [0] 
[Fpart-Bounded(Base)
] , \linebreak [0] 
[Fpart-Bounded(InduHelper)
] , \linebreak [0] 
[Fpart-Bounded(Indu)
] , \linebreak [0] 
[Fpart-Bounded
] , \linebreak [0] 
[F-Bounded(Helper)
] , \linebreak [0] 
[F-Bounded
] , \linebreak [0] 
[SameFmultiplication(Helper)
] , \linebreak [0] 
[SameFmultiplication
] , \linebreak [0] 
[EqMultiplication(R)
] , \linebreak [0] 
[EqMultiplicationLeft(R)
] , \linebreak [0] 
[x*0=0(F)
] , \linebreak [0] 
[x*0=0(R)
] , \linebreak [0] 
[LessMultiplication(F)(Helper2)
] , \linebreak [0] 
[LessMultiplication(F)(Helper)
] , \linebreak [0] 
[LessMultiplication(F)
] , \linebreak [0] 
[LessMultiplication(R)
] , \linebreak [0] 
[LeqMultiplication(R)
] , \linebreak [0] 
[PlusAssociativity(F)
] , \linebreak [0] 
[Plus0(F)
] , \linebreak [0] 
[PlusCommutativity(F)
] , \linebreak [0] 
[TimesAssociativity(F)
] , \linebreak [0] 
[Times1f
] , \linebreak [0] 
[Cauchy(2)(Helper)
] , \linebreak [0] 
[Cauchy(2)
] , \linebreak [0] 
[ReciprocalFnonzero
] , \linebreak [0] 
[(Eventually=f)2sameF(Helper)
] , \linebreak [0] 
[(Eventually=f)2sameF
] , \linebreak [0] 
[FromNotSameF(Strong)(Helper2)
] , \linebreak [0] 
[FromNotSameF(Strong)(Helper)
] , \linebreak [0] 
[FromNotSameF(Strong)
] , \linebreak [0] 
[SameFreciprocal(Helper)
] , \linebreak [0] 
[SameFreciprocal
] , \linebreak [0] 
[From!!==
] , \linebreak [0] 
[Reciprocal(R)
] , \linebreak [0] 
[TimesCommutativity(F)
] , \linebreak [0] 
[Distribution(F)
] , \linebreak [0] 
[FromMax(1)
] , \linebreak [0] 
[FromMax(2)
] , \linebreak [0] 
[ToNegatedAnd
] , \linebreak [0] 
[DistributionOut
] , \linebreak [0] 
[DistributionOutLeft
] , \linebreak [0] 
[DistributionLeft
] , \linebreak [0] 
[FromNotLess(R)
] , \linebreak [0] 
[CartProdIsRelation
] , \linebreak [0] 
[FromSubset
] , \linebreak [0] 
[SubsetIsRelation
] , \linebreak [0] 
[ToSeries
] , \linebreak [0] 
[FromSeries
] , \linebreak [0] 
[SeriesSubsetCP
] , \linebreak [0] 
[ValueType
] , \linebreak [0] 
[RemoveOr
] , \linebreak [0] 
[FromSingleton
] , \linebreak [0] 
[InPair(1)
] , \linebreak [0] 
[InPair(2)
] , \linebreak [0] 
[SameMember(2)
] , \linebreak [0] 
[ToBinaryUnion(1)
] , \linebreak [0] 
[ToBinaryUnion(2)
] , \linebreak [0] 
[FromOrderedPair(TwoLevels)
] , \linebreak [0] 
[ToCartProd(Helper)
] , \linebreak [0] 
[ToCartProd
] , \linebreak [0] 
[NonreciprocalToRight(Eq)
] , \linebreak [0] 
[NonreciprocalToLeft(Eq)(1 term)
] , \linebreak [0] 
[SameReciprocal
] , \linebreak [0] 
[CPseparationIsRelation
] , \linebreak [0] 
[OrderedPairEquality
] , \linebreak [0] 
[ReciprocalIsFunction
] , \linebreak [0] 
[ReciprocalIsTotal
] , \linebreak [0] 
[ReciprocalIsRationalSeries
] , \linebreak [0] 
[CrsIsRelation
] , \linebreak [0] 
[CrsIsFunction 
] , \linebreak [0] 
[CrsIsTotal
] , \linebreak [0] 
[CrsIsSeries
] , \linebreak [0] 
[CrsLookup
] , \linebreak [0] 
[0f
] , \linebreak [0] 
[1f
] , \linebreak [0] 
[ToSingleton
] , \linebreak [0] 
[FromSameSingleton
] , \linebreak [0] 
[SingletonmembersEqual
] , \linebreak [0] 
[UnequalsNotInSingleton
] , \linebreak [0] 
[NonsingletonmembersUnequal
] , \linebreak [0] 
[FromOrderedPair
] , \linebreak [0] 
[FromOrderedPair(1)
] , \linebreak [0] 
[FromOrderedPair(2)
] , \linebreak [0] 
[FromCartProd
] , \linebreak [0] 
[FromCartProd(1)
] , \linebreak [0] 
[FromCartProd(2)
] , \linebreak [0] 
[sameOrderedPair
] , \linebreak [0] 
[InSeriesHelper
] , \linebreak [0] 
[InSeries
] , \linebreak [0] 
[To=f(Subset)(Helper)
] , \linebreak [0] 
[To=f(Subset)
] , \linebreak [0] 
[To=f
] , \linebreak [0] 
[productIsFunction
] , \linebreak [0] 
[productIsTotal
] , \linebreak [0] 
[ProductIsRationalSeries
] , \linebreak [0] 
[TimesF
] , \linebreak [0] 
[-x+(1/2)x=-(1/2)x
] , \linebreak [0] 
[PositiveTripled
] , \linebreak [0] 
[PositiveDividedBy3
] , \linebreak [0] 
[|x-x|=0
] , \linebreak [0] 
[1<2
] , \linebreak [0] 
[1/3<2/3
] , \linebreak [0] 
[(1/3)x+(1/3)x=(2/3)x
] , \linebreak [0] 
[(2/3)x+(1/3)x=x
] , \linebreak [0] 
[-x+(2/3)x=-(1/3)x
] , \linebreak [0] 
[-(1/3)x-(1/3)x=-(2/3)x
] , \linebreak [0] 
[-x+(1/3)x=-(2/3)x
] , \linebreak [0] 
[PreserveLessGreater
] , \linebreak [0] 
[ClosetolessIsLess
] , \linebreak [0] 
[SubLessLeft(F)
] , \linebreak [0] 
[SubLessLeft(R)
] , \linebreak [0] 
[ClosetogreaterIsGreater
] , \linebreak [0] 
[SubLessRight(F)
] , \linebreak [0] 
[SubLessRight(R)
] , \linebreak [0] 
[plus0Left
] , \linebreak [0] 
[times1Left
] , \linebreak [0] 
[EqAdditionLeft
] , \linebreak [0] 
[EqMultiplicationLeft
] , \linebreak [0] 
[PlusF(Sym)
] , \linebreak [0] 
[TimesF(Sym)
] , \linebreak [0] 
[SameSeries(Gen)
] , \linebreak [0] 
[EqualsSameF
] , \linebreak [0] 
[LeqReflexivity(R)
] , \linebreak [0] 
[LeqTotality(R)
] , \linebreak [0] 
[PositiveToLeft(Eq)
] , \linebreak [0] 
[ExpZero(Exact) 
] , \linebreak [0] 
[SameExp(Base)
] , \linebreak [0] 
[SameExp(Indu)
] , \linebreak [0] 
[SameExp
] , \linebreak [0] 
[(1/2)(x+y)-x=(1/2)(y-x)
] , \linebreak [0] 
[y-(1/2)(x+y)=(1/2)(y-x)
] , \linebreak [0] 
[BSzero(Exact)
] , \linebreak [0] 
[SameBS(2)(Base)
] , \linebreak [0] 
[SameBS(2)(Indu)
] , \linebreak [0] 
[SameBS(2)
] , \linebreak [0] 
[NegativeToLeft(Less)(1 term)
] , \linebreak [0] 
[BS(+1)
] , \linebreak [0] 
[BSbound(Exact)(Base)
] , \linebreak [0] 
[BSbound(Exact)(Indu)
] , \linebreak [0] 
[BSbound(Exact)
] , \linebreak [0] 
[BSbound
] , \linebreak [0] 
[UStelescope(Zero)(Exact)
] , \linebreak [0] 
[SameTelescope(2)(Base)
] , \linebreak [0] 
[SameTelescope(2)(Indu)
] , \linebreak [0] 
[SameTelescope(2)
] , \linebreak [0] 
[Exp(+1)
] , \linebreak [0] 
[PositiveBase(Base)
] , \linebreak [0] 
[PositiveBase(Indu)
] , \linebreak [0] 
[PositiveBase
] , \linebreak [0] 
[TelescopeNumerical(Base)
] , \linebreak [0] 
[TelescopeNumerical(Indu)
] , \linebreak [0] 
[TelescopeNumerical
] , \linebreak [0] 
[(+1)IsPositive(N)
] , \linebreak [0] 
[DistributionOut(Minus)
] , \linebreak [0] 
[PositiveToRight(Eq)(1 term)
] , \linebreak [0] 
[SameSeries(NumDiff)
] , \linebreak [0] 
[ToNegatedDoubleImply
] , \linebreak [0] 
[AddNegatedAll
] , \linebreak [0] 
[(A)to(~E~)(Imply)
] , \linebreak [0] 
[(E)to(~A~)(Imply)
] , \linebreak [0] 
[(E~)to(~A)(Imply)
] , \linebreak [0] 
[ToNegatedAEA 
] , \linebreak [0] 
[Three2threeTerms(R)
] , \linebreak [0] 
[LessNeq(F)(Helper)
] , \linebreak [0] 
[LessNeq(F)
] , \linebreak [0] 
[LessNeq(R)
] , \linebreak [0] 
[x=x+(y-y)(R)
] , \linebreak [0] 
[x=x+y-y(R)
] , \linebreak [0] 
[SubtractEquations(R)
] , \linebreak [0] 
[NeqAddition(R)
] , \linebreak [0] 
[PositiveToRight(Less)(R)
] , \linebreak [0] 
[PositiveToRight(Less)(1 term)(R)
] , \linebreak [0] 
[LeqNeqLess(R)
] , \linebreak [0] 
[SubLeqLeft(R)
] , \linebreak [0] 
[ToLeq(Advanced)(R)
] , \linebreak [0] 
[LeqLessTransitivity(R)
] , \linebreak [0] 
[NegativeToLeft(Eq)(R)
] , \linebreak [0] 
[NegativeToRight(Less)(R)
] , \linebreak [0] 
[!!==Symmetry
] , \linebreak [0] 
[SwitchTerms(x<=y-z)
] , \linebreak [0] 
[Plus0Left(R)
] , \linebreak [0] 
[PositiveToRight(Eq)(R)
] , \linebreak [0] 
[EqAdditionLeft(R)
] , \linebreak [0] 
[Three2twoTerms(R)
] , \linebreak [0] 
[To!!==
] , \linebreak [0] 
[PositiveToRight(Less)(1 term)
] , \linebreak [0] 
[(A~)to(~E)
] , \linebreak [0] 
[NegativeToRight(Eq)(R)
] , \linebreak [0] 
[NegativeToRight(Eq)(1 term)(R)
] , \linebreak [0] 
[DoubleMinus(R)
] , \linebreak [0] 
[UniqueNegative(R)
] , \linebreak [0] 
[SubtractEquationsLeft(R)
] , \linebreak [0] 
[EqNegated(R)
] , \linebreak [0] 
[NeqNegated(R)
] , \linebreak [0] 
[-0=0(R)
] , \linebreak [0] 
[NegativeNegated(R)
] , \linebreak [0] 
[FromLeqGeq(R)
] , \linebreak [0] 
[0<=|x|(R)
] , \linebreak [0] 
[PositiveNegated(R)
] , \linebreak [0] 
[AddEquations(R)
] , \linebreak [0] 
[Times(-1)(R)
] , \linebreak [0] 
[Times(-1)Left(R)
] , \linebreak [0] 
[-x-y=-(x+y)(R)
] , \linebreak [0] 
[LessTotality(R)
] , \linebreak [0] 
[SameNumerical(R)
] , \linebreak [0] 
[MinusNegated(R)
] , \linebreak [0] 
[PositiveNumerical(R)
] , \linebreak [0] 
[SignNumerical(+)(R)
] , \linebreak [0] 
[NonnegativeNumerical(R)
] , \linebreak [0] 
[NegativeNumerical(R)
] , \linebreak [0] 
[LeqNegated(R)
] , \linebreak [0] 
[LessNegated(R)
] , \linebreak [0] 
[SubLeqRight(R)
] , \linebreak [0] 
[FromLess(R)
] , \linebreak [0] 
[DistributionOut(R)
] , \linebreak [0] 
[x*0+x=x(R)
] , \linebreak [0] 
[x*0=0(R)(fff)
] , \linebreak [0] 
[SignNumerical(R)
] , \linebreak [0] 
[NumericalDifference(R)
] , \linebreak [0] 
[x<=|x|(R)
] , \linebreak [0] 
[USlimitIsUpperBound(Helper)
] , \linebreak [0] 
[USlimitIsUpperBound
] , \linebreak [0] 
[(-1)*(-1)+(-1)*1=0(R)
] , \linebreak [0] 
[(-1)*(-1)=1(R)
] , \linebreak [0] 
[0<1Helper(R)
] , \linebreak [0] 
[0<1(R)
] , \linebreak [0] 
[ExpZero(Exact)(R)
] , \linebreak [0] 
[PositiveBase(R)(Base)
] , \linebreak [0] 
[Three2twoFactors(R)
] , \linebreak [0] 
[x=x*y*(1/y)(R)
] , \linebreak [0] 
[NeqMultiplication(R)
] , \linebreak [0] 
[LessTransitivity(R)
] , \linebreak [0] 
[0<2(R)
] , \linebreak [0] 
[SameExp(R)(Base)
] , \linebreak [0] 
[SameExp(R)(Indu)
] , \linebreak [0] 
[SameExp(R)
] , \linebreak [0] 
[SubNeqLeft(R)
] , \linebreak [0] 
[SubNeqRight(R)
] , \linebreak [0] 
[NonzeroFactors(R)
] , \linebreak [0] 
[NonnegativeFactors(R)
] , \linebreak [0] 
[PositiveFactors(R)
] , \linebreak [0] 
[LessDivision(R)
] , \linebreak [0] 
[0<1/2(R)
] , \linebreak [0] 
[PositiveToRight(Eq)(1 term)(R)
] , \linebreak [0] 
[Exp(+1)(R)
] , \linebreak [0] 
[PositiveBase(R)(Indu)
] , \linebreak [0] 
[PositiveBase(R)
] , \linebreak [0] 
[-x*y=-(x*y)(R)
] , \linebreak [0] 
[PositiveToLeft(Eq)(R)
] , \linebreak [0] 
[Times1Left(R)
] , \linebreak [0] 
[x+x=2*x(R)
] , \linebreak [0] 
[(1/2)x+(1/2)x=x(R)
] , \linebreak [0] 
[DistributionOut(Minus)(R)
] , \linebreak [0] 
[(1/2)(x+y)-x=(1/2)(y-x)(R)
] , \linebreak [0] 
[IntervalSize(R)(Base)
] , \linebreak [0] 
[LessMultiplicationLeft(R)
] , \linebreak [0] 
[NegativeToLeft(Less)(R)
] , \linebreak [0] 
[NegativeToLeft(Less)(1 term)(R)
] , \linebreak [0] 
[y-(1/2)(x+y)=(1/2)(y-x)(R)
] , \linebreak [0] 
[IntervalSize(R)(Indu)
] , \linebreak [0] 
[IntervalSize(R)
] , \linebreak [0] 
[XSlessUS(R)
] , \linebreak [0] 
[USdecreasing(+1)(R)
] , \linebreak [0] 
[ExpUnbounded(Base)
] , \linebreak [0] 
[ExpUnbounded(Indu)
] , \linebreak [0] 
[ExpUnbounded
] , \linebreak [0] 
[1<=x+1(N)
] , \linebreak [0] 
[ExpNonzero(Base)
] , \linebreak [0] 
[ExpNonzero(Indu)
] , \linebreak [0] 
[ExpNonzero
] , \linebreak [0] 
[ExpNonzero(2)
] , \linebreak [0] 
[HalfBase(Base)
] , \linebreak [0] 
[HalfBase(Indu)
] , \linebreak [0] 
[MultiplyEquations(R)
] , \linebreak [0] 
[NonreciprocalToRight(Eq)(1 term)(R)
] , \linebreak [0] 
[PositiveNonzero(R)
] , \linebreak [0] 
[NonzeroProduct(2)(R)
] , \linebreak [0] 
[HalfBase
] , \linebreak [0] 
[Three2threeFactors(R)
] , \linebreak [0] 
[x*y=zBackwards(R)
] , \linebreak [0] 
[PositiveInverted(R)
] , \linebreak [0] 
[ReciprocalToRight(Less)(R)
] , \linebreak [0] 
[ReciprocalToRight(Less)(1 term)(R)
] , \linebreak [0] 
[NonreciprocalToLeft(Less)(R)
] , \linebreak [0] 
[1<x*y(R)
] , \linebreak [0] 
[SwitchFactors(1/x<y)(R)
] , \linebreak [0] 
[SmallHalving
] , \linebreak [0] 
[IntervalSize(anyPositive)
] , \linebreak [0] 
[USdecreasing(+n)(Base)
] , \linebreak [0] 
[USdecreasing(+n)(Indu)
] , \linebreak [0] 
[USdecreasing(+n)
] , \linebreak [0] 
[USdecreasing
] , \linebreak [0] 
[LeqAdditionLeft(R)
] , \linebreak [0] 
[ToNotLess(R)
] , \linebreak [0] 
[LimitOfUSIsLeq
] , \linebreak [0] 
[SubtractEquations(Less)(R)
] , \linebreak [0] 
[SubtractEquationsLeft(Less)(R)
] , \linebreak [0] 
[LessNegated(Negative)(R)
] , \linebreak [0] 
[FromNegatedAnd(Imply)
] , \linebreak [0] 
[RemoveDoubleNeg(Consequent)
] , \linebreak [0] 
[FromNotUpperBound
] , \linebreak [0] 
[LeqNUB
] , \linebreak [0] 
[USlimitIsLeastUpperBound(Helper)
] , \linebreak [0] 
[USlimitIsLeastUpperBound
] , \linebreak [0] 
[ExistMP3
] , \linebreak [0] 
[GreaterPositive(N)
] , \linebreak [0] 
[ysFClose(Helper)
] , \linebreak [0] 
[ysFClose
] , \linebreak [0] 
[ysFCauchy(Helper)
] , \linebreak [0] 
[ysFCauchy
] , \linebreak [0] 
[from<<==
] , \linebreak [0] 
[to<<==
] , \linebreak [0] 
[NonnegativeNumerical(F)
] , \linebreak [0] 
[NegativeNumerical(F)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\mbox{tester1}
] , \linebreak [0] 
[
\mbox{tester2}
] , \linebreak [0] 
[
\mbox{tester3}
] , \linebreak [0] 
[
\mbox{tester4}
] , \linebreak [0] 
[
\mbox{tester5}
] , \linebreak [0] 
[
\mbox{tester6}
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\_\{
\ast 
\}
] , \linebreak [0] 
[
\ast 
/indexintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/bothintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/nameintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast '
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\Rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
0
] , \linebreak [0] 
[
\ast 
1
] , \linebreak [0] 
[
0 \mathrm {b}
] , \linebreak [0] 
[
\ast 
\mbox {-color}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
\mbox {-color}^{\ast}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
{}^H
] , \linebreak [0] 
[
\ast 
{}^T
] , \linebreak [0] 
[
\ast 
{}^U
] , \linebreak [0] 
[
\ast 
{}^h
] , \linebreak [0] 
[
\ast 
{}^t
] , \linebreak [0] 
[
\ast 
{}^s
] , \linebreak [0] 
[
\ast 
{}^c
] , \linebreak [0] 
[
\ast 
{}^d
] , \linebreak [0] 
[
\ast 
{}^a
] , \linebreak [0] 
[
\ast 
{}^C
] , \linebreak [0] 
[
\ast 
{}^M
] , \linebreak [0] 
[
\ast 
{}^B
] , \linebreak [0] 
[
\ast 
{}^{r}
] , \linebreak [0] 
[
\ast 
{}^{i}
] , \linebreak [0] 
[
\ast 
{}^{d}
] , \linebreak [0] 
[
\ast 
{}^{R}
] , \linebreak [0] 
[
\ast 
{}^{0}
] , \linebreak [0] 
[
\ast 
{}^{1}
] , \linebreak [0] 
[
\ast 
{}^{2}
] , \linebreak [0] 
[
\ast 
{}^{3}
] , \linebreak [0] 
[
\ast 
{}^{4}
] , \linebreak [0] 
[
\ast 
{}^{5}
] , \linebreak [0] 
[
\ast 
{}^{6}
] , \linebreak [0] 
[
\ast 
{}^{7}
] , \linebreak [0] 
[
\ast 
{}^{8}
] , \linebreak [0] 
[
\ast 
{}^{9}
] , \linebreak [0] 
[
\ast 
{} ^ { E }
] , \linebreak [0] 
[
\ast 
{} ^ {\cal V}
] , \linebreak [0] 
[
\ast 
{} ^ {\cal C}
] , \linebreak [0] 
[
\ast 
{} ^ {{\cal C} ^ { \ast }}
] , \linebreak [0] 
[
\ast 
{}^{hide}
]
; 
\newline \mathbf {Preassociative} \newline 
[
\mbox{``}
\ast 
\mbox{''}
] , \linebreak [0] 
[
] , \linebreak [0] 
[
(
\ast 
)^{\bf t}
] , \linebreak [0] 
[
\mbox{string}(
\ast 
)+
\ast 
] , \linebreak [0] 
[\mbox{string}(
\ast 
)\mathrel{++}
\ast 
] , \linebreak [0] 
[
\newline 
\ast 
] , \linebreak [0] 
[
\linebreak [0]\ \hskip0em plus2.0em{}
\ast 
] , \linebreak [0] 
[!
\ast 
] , \linebreak [0] 
[
\mbox {\tt \char34}
\ast 
] , \linebreak [0] 
[
\#
\ast 
] , \linebreak [0] 
[
\$
\ast 
] , \linebreak [0] 
[
\%
\ast 
] , \linebreak [0] 
[
\&
\ast 
] , \linebreak [0] 
[
\mbox {'}
\ast 
] , \linebreak [0] 
[(
\ast 
] , \linebreak [0] 
[)
\ast 
] , \linebreak [0] 
[
{*}
\ast 
] , \linebreak [0] 
[
{+}
\ast 
] , \linebreak [0] 
[,
\ast 
] , \linebreak [0] 
[
\mbox{-}
\ast 
] , \linebreak [0] 
[.
\ast 
] , \linebreak [0] 
[/
\ast 
] , \linebreak [0] 
[0
\ast 
] , \linebreak [0] 
[1
\ast 
] , \linebreak [0] 
[2
\ast 
] , \linebreak [0] 
[3
\ast 
] , \linebreak [0] 
[4
\ast 
] , \linebreak [0] 
[5
\ast 
] , \linebreak [0] 
[6
\ast 
] , \linebreak [0] 
[7
\ast 
] , \linebreak [0] 
[8
\ast 
] , \linebreak [0] 
[9
\ast 
] , \linebreak [0] 
[
{:}
\ast 
] , \linebreak [0] 
[;
\ast 
] , \linebreak [0] 
[
{<}
\ast 
] , \linebreak [0] 
[
{=}
\ast 
] , \linebreak [0] 
[
{>}
\ast 
] , \linebreak [0] 
[?
\ast 
] , \linebreak [0] 
[@
\ast 
] , \linebreak [0] 
[A
\ast 
] , \linebreak [0] 
[B
\ast 
] , \linebreak [0] 
[C
\ast 
] , \linebreak [0] 
[D
\ast 
] , \linebreak [0] 
[E
\ast 
] , \linebreak [0] 
[F
\ast 
] , \linebreak [0] 
[G
\ast 
] , \linebreak [0] 
[H
\ast 
] , \linebreak [0] 
[I
\ast 
] , \linebreak [0] 
[J
\ast 
] , \linebreak [0] 
[K
\ast 
] , \linebreak [0] 
[L
\ast 
] , \linebreak [0] 
[M
\ast 
] , \linebreak [0] 
[N
\ast 
] , \linebreak [0] 
[O
\ast 
] , \linebreak [0] 
[P
\ast 
] , \linebreak [0] 
[Q
\ast 
] , \linebreak [0] 
[R
\ast 
] , \linebreak [0] 
[S
\ast 
] , \linebreak [0] 
[T
\ast 
] , \linebreak [0] 
[U
\ast 
] , \linebreak [0] 
[V
\ast 
] , \linebreak [0] 
[W
\ast 
] , \linebreak [0] 
[X
\ast 
] , \linebreak [0] 
[Y
\ast 
] , \linebreak [0] 
[Z
\ast 
] , \linebreak [0] 
[[
\ast 
] , \linebreak [0] 
[
\mbox {$\backslash $}
\ast 
] , \linebreak [0] 
[]
\ast 
] , \linebreak [0] 
[
{\char94}
\ast 
] , \linebreak [0] 
[
\_
\ast 
] , \linebreak [0] 
[
\mbox {`}
\ast 
] , \linebreak [0] 
[a
\ast 
] , \linebreak [0] 
[b
\ast 
] , \linebreak [0] 
[c
\ast 
] , \linebreak [0] 
[d
\ast 
] , \linebreak [0] 
[e
\ast 
] , \linebreak [0] 
[f
\ast 
] , \linebreak [0] 
[g
\ast 
] , \linebreak [0] 
[h
\ast 
] , \linebreak [0] 
[i
\ast 
] , \linebreak [0] 
[j
\ast 
] , \linebreak [0] 
[k
\ast 
] , \linebreak [0] 
[l
\ast 
] , \linebreak [0] 
[m
\ast 
] , \linebreak [0] 
[n
\ast 
] , \linebreak [0] 
[o
\ast 
] , \linebreak [0] 
[p
\ast 
] , \linebreak [0] 
[q
\ast 
] , \linebreak [0] 
[r
\ast 
] , \linebreak [0] 
[s
\ast 
] , \linebreak [0] 
[t
\ast 
] , \linebreak [0] 
[u
\ast 
] , \linebreak [0] 
[v
\ast 
] , \linebreak [0] 
[w
\ast 
] , \linebreak [0] 
[x
\ast 
] , \linebreak [0] 
[y
\ast 
] , \linebreak [0] 
[z
\ast 
] , \linebreak [0] 
[
\{
\ast 
] , \linebreak [0] 
[|
\ast 
] , \linebreak [0] 
[
\}
\ast 
] , \linebreak [0] 
[
\char126 
\ast 
] , \linebreak [0] 
[
\mathbf{Preassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
\mathbf{Postassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
[
\ast 
] , \linebreak [0] 
\ast 
] , \linebreak [0] 
[
\mathrm{priority} \,
\ast 
\, \mathrm{end}
] , \linebreak [0] 
[
newline\ 
\ast 
] , \linebreak [0] 
[
macro\ newline\ 
\ast 
] , \linebreak [0] 
[
MacroIndent(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbin {\mbox {'}}
\ast 
] , \linebreak [0] 
[
\ast 
\mathbin {\mbox {`}}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[ 
\ast 
 (exp) 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{}'
] , \linebreak [0] 
[R(
\ast 
)
] , \linebreak [0] 
[--R(
\ast 
)
] , \linebreak [0] 
[rec
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 / 
\ast 
] , \linebreak [0] 
[
\ast 
 \cap 
\ast 
] , \linebreak [0] 
[
\ast 
[
\ast 
]
]
; 
\newline \mathbf {Preassociative} \newline 
[\cup 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\cup} 
\ast 
] , \linebreak [0] 
[P(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[\{
\ast 
\}
] , \linebreak [0] 
[StateExpand(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[extractSeries(
\ast 
)
] , \linebreak [0] 
[SetOfSeries(
\ast 
)
] , \linebreak [0] 
[--Macro(
\ast 
)
] , \linebreak [0] 
[ExpandList(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[**Macro(
\ast 
)
] , \linebreak [0] 
[++Macro(
\ast 
)
] , \linebreak [0] 
[<<Macro(
\ast 
)
] , \linebreak [0] 
[||Macro(
\ast 
)
] , \linebreak [0] 
[01//Macro(
\ast 
)
] , \linebreak [0] 
[UB(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[LUB(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[BS(
\ast 
,
\ast 
)
] , \linebreak [0] 
[UStelescope(
\ast 
,
\ast 
)
] , \linebreak [0] 
[(
\ast 
)
] , \linebreak [0] 
[|f
\ast 
|
] , \linebreak [0] 
[|r
\ast 
|
] , \linebreak [0] 
[Limit(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Union(
\ast 
)
] , \linebreak [0] 
[IsOrderedPair(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsRelation(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[isFunction(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsSeries(
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsNatural(
\ast 
,\ast
)
] , \linebreak [0] 
[OrderedPair(
\ast 
,
\ast 
)
] , \linebreak [0] 
[TypeNat(
\ast 
)
] , \linebreak [0] 
[TypeNat0(
\ast 
)
] , \linebreak [0] 
[TypeRational(
\ast 
)
] , \linebreak [0] 
[TypeRational0(
\ast 
)
] , \linebreak [0] 
[TypeSeries(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Typeseries0(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[\{
\ast 
,
\ast 
\}
] , \linebreak [0] 
[\langle 
\ast 
,
\ast 
\rangle
] , \linebreak [0] 
[(-u
\ast 
)
] , \linebreak [0] 
[-_{f}
\ast 
] , \linebreak [0] 
[(--
\ast 
)
] , \linebreak [0] 
[1f/
\ast 
] , \linebreak [0] 
[01//temp
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
(
\ast 
,
\ast 
)
] , \linebreak [0] 
[ReflRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[SymRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[TransRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[EqRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[[
\ast 
 \mathrel{\in} 
\ast 
]_{
\ast 
}
] , \linebreak [0] 
[Partition(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cdot 
\ast 
] , \linebreak [0] 
[
\ast 
\cdot_0 
\ast 
] , \linebreak [0] 
[(
\ast 
*
\ast 
)
] , \linebreak [0] 
[
\ast 
*_{f}
\ast 
] , \linebreak [0] 
[
\ast 
**
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
+ 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_1} 
\ast 
] , \linebreak [0] 
[
\ast 
- 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_1} 
\ast 
] , \linebreak [0] 
[(
\ast 
+
\ast 
)
] , \linebreak [0] 
[(
\ast 
-
\ast 
)
] , \linebreak [0] 
[
\ast 
+_{f}
\ast 
] , \linebreak [0] 
[
\ast 
-_{f}
\ast 
] , \linebreak [0] 
[
\ast 
++
\ast 
] , \linebreak [0] 
[R(
\ast 
) -- R(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 \mathrel{\in} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[|
\ast 
|
] , \linebreak [0] 
[if(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[Max(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Max(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 = 
\ast 
] , \linebreak [0] 
[
\ast 
 \neq 
\ast 
] , \linebreak [0] 
[
\ast 
 <= 
\ast 
] , \linebreak [0] 
[
\ast 
 < 
\ast 
] , \linebreak [0] 
[
\ast 
<_{f}
\ast 
] , \linebreak [0] 
[
\ast 
\leq_{f}
\ast 
] , \linebreak [0] 
[SF(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
 == 
\ast 
] , \linebreak [0] 
[
\ast 
 !!== 
\ast 
] , \linebreak [0] 
[
\ast 
 << 
\ast 
] , \linebreak [0] 
[
\ast 
 <<== 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cup \{ 
\ast 
\}
] , \linebreak [0] 
[
\ast 
\cup 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \{ 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel { \dot { . \, . } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { \dot { . \, . } } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { : \, : } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { {+} 2 \ast } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { : \, : }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { {+} 2 \ast }
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
,\linebreak [0] 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\stackrel {B}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {D}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {C}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {P}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\approx 
\ast 
] , \linebreak [0] 
[
\ast 
=
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {+}{\rightarrow }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t^*}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{r}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_t 
\ast 
] , \linebreak [0] 
[
\ast 
\subseteq_T 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{T}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{s}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in}^{\ast} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for}^{\ast} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_c 
\ast 
] , \linebreak [0] 
[
\ast 
< 
\ast 
] , \linebreak [0] 
[
\ast 
<' 
\ast 
] , \linebreak [0] 
[
\ast 
\le' 
\ast 
] , \linebreak [0] 
[
\ast 
= 
\ast 
] , \linebreak [0] 
[
\ast 
\neq 
\ast 
] , \linebreak [0] 
[
\ast 
{}^{var}
] , \linebreak [0] 
[
\ast 
\#^0
\ast 
] , \linebreak [0] 
[
\ast 
\#^1
\ast 
] , \linebreak [0] 
[
\ast 
\#^*
\ast 
] , \linebreak [0] 
[
\ast 
\!\mathrel{==}\! 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\subseteq} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
{\neg }
\ast 
] , \linebreak [0] 
[\dot{\neg}\,(
\ast 
)n
] , \linebreak [0] 
[
\ast 
 \mathrel{\notin} 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\neq} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\wedge 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\tilde{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\wedge_c 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\dot{\wedge}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\vee 
\ast 
] , \linebreak [0] 
[
\ast 
\parallel 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\vee}} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
 \mathrel{\dot{\vee}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\exists 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall_{obj} 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\exists 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\ddot{\Rightarrow}} 
\ast 
] , \linebreak [0] 
[
\ast 
\Rightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\Leftrightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\dot{\Leftrightarrow}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[ \{ ph \mathrel{\in} 
\ast 
\mid 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
:
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{spy}
\ast 
] , \linebreak [0] 
[
\ast 
!
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\left\{\protect \begin {array}{l}
\ast 
\\
\ast 
\protect \end {array}\right.
]
; 
\newline \mathbf {Preassociative} \newline 
[
\lambda 
\ast 
.
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
. 
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
] , \linebreak [0] 
[
{\bf if} \ 
\ast 
\ {\bf then} \ 
\ast 
\ {\bf else} \ 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
= 
\ast 
\mathbf{\ in\ } 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
\mathrel{\ddot{=}} 
\ast 
\mathrel{\ in\ } 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\#
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{} ^ { I }
] , \linebreak [0] 
[
\ast 
{} ^ { \rhd }
] , \linebreak [0] 
[
\ast 
{} ^ { V } 
] , \linebreak [0] 
[
\ast 
{} ^ { + } 
] , \linebreak [0] 
[
\ast 
{} ^ { - } 
] , \linebreak [0] 
[
\ast 
{} ^ { \ast } 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathop {\char64} 
\ast 
] , \linebreak [0] 
[
\ast 
\rhd 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\ast 
] , \linebreak [0] 
[
\ast 
\gg 
\ast 
] , \linebreak [0] 
[
\ast 
\unrhd 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\vdash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {i.e.} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\Pi 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel {\oplus} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
; 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\ proves\ 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbf{\ proof\ of\ } 
\ast 
: 
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: 
\ast 
\gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Last\ line \, 
\ast 
\gg 
\ast 
\,\Box
] , \linebreak [0] 
[
Line \, 
\ast 
: Premise \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: \mbox{Side-condition} \gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Local \gg 
\ast 
= 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Begin \, 
\ast 
 ; 
\ast 
 : End ; 
\ast 
] , \linebreak [0] 
[
Last\ block\ line \, 
\ast 
\gg 
\ast 
\,;
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{|} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\mathrm{,}}
\ast 
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\& 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\backslash \backslash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{linebreak[4]} 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \backslash 
\ast 
]
; 
\mathbf{End\ table}$
\end {flushleft}

\section{Pyk definitioner} \label{sec:pyk}

\begin{flushleft}
$(
[LeqTotality(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqTotality(R)
\mbox{''}
]
\linebreak[4] 
[PositiveToLeft(Eq)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToLeft(Eq)
\mbox{''}
]
\linebreak[4] 
[ExpZero(Exact) 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expZero
\linebreak [0]\ \hskip0em plus2.0em{}exact
\mbox{''}
]
\linebreak[4] 
[SameExp(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[SameExp(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[SameExp
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp
\mbox{''}
]
\linebreak[4] 
[(1/2)(x+y)-x=(1/2)(y-x)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(1/2)(x
{+}y)
\mbox{-}x
{=}(1/2)(y
\mbox{-}x)
\mbox{''}
]
\linebreak[4] 
[y-(1/2)(x+y)=(1/2)(y-x)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}y
\mbox{-}(1/2)(x
{+}y)
{=}(1/2)(y
\mbox{-}x)
\mbox{''}
]
\linebreak[4] 
[BSzero(Exact)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}exact
\mbox{''}
]
\linebreak[4] 
[SameBS(2)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameBase(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[SameBS(2)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameBase(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[SameBS(2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameBase(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}second
\mbox{''}
]
\linebreak[4] 
[NegativeToLeft(Less)(1 term)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToLeft(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]
\linebreak[4] 
[BS(+1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum(
{+}1)
\mbox{''}
]
\linebreak[4] 
[BSbound(Exact)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}exact
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[BSbound(Exact)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}exact
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[BSbound(Exact)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}exact
\linebreak [0]\ \hskip0em plus2.0em{}bound
\mbox{''}
]
\linebreak[4] 
[BSbound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}bound
\mbox{''}
]
\linebreak[4] 
[UStelescope(Zero)(Exact)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}UStelescope
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}exact
\mbox{''}
]
\linebreak[4] 
[SameTelescope(2)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameTelescope
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[SameTelescope(2)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameTelescope
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[SameTelescope(2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameTelescope
\linebreak [0]\ \hskip0em plus2.0em{}second
\mbox{''}
]
\linebreak[4] 
[Exp(+1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}exp(
{+}1)
\mbox{''}
]
\linebreak[4] 
[PositiveBase(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[PositiveBase(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[PositiveBase
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase
\mbox{''}
]
\linebreak[4] 
[TelescopeNumerical(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}telescopeNumerical
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[TelescopeNumerical(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}telescopeNumerical
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[TelescopeNumerical
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}telescopeNumerical
\mbox{''}
]
\linebreak[4] 
[(+1)IsPositive(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{+}1IsPositive(N)
\mbox{''}
]
\linebreak[4] 
[DistributionOut(Minus)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}distributionOut(Minus)
\mbox{''}
]
\linebreak[4] 
[PositiveToRight(Eq)(1 term)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]
\linebreak[4] 
[SameSeries(NumDiff)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameSeries(NumDiff)
\mbox{''}
]
\linebreak[4] 
[ToNegatedDoubleImply
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}to
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}imply
\mbox{''}
]
\linebreak[4] 
[AddNegatedAll
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}addNegatedAll
\mbox{''}
]
\linebreak[4] 
[(A)to(~E~)(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(A)to(
\char126 E
\char126 )(Imply)
\mbox{''}
]
\linebreak[4] 
[(E)to(~A~)(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(E)to(
\char126 A
\char126 )(Imply)
\mbox{''}
]
\linebreak[4] 
[(E~)to(~A)(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(E
\char126 )to(
\char126 A)(Imply)
\mbox{''}
]
\linebreak[4] 
[ToNegatedAEA 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}toNegatedAEA
\mbox{''}
]
\linebreak[4] 
[Three2threeTerms(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}three2threeTerms(R)
\mbox{''}
]
\linebreak[4] 
[LessNeq(F)(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNeq(F)
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]
\linebreak[4] 
[LessNeq(F)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNeq(F)
\mbox{''}
]
\linebreak[4] 
[LessNeq(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNeq(R)
\mbox{''}
]
\linebreak[4] 
[x=x+(y-y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{=}x
{+}(y
\mbox{-}y)(R)
\mbox{''}
]
\linebreak[4] 
[x=x+y-y(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{=}x
{+}y
\mbox{-}y(R)
\mbox{''}
]
\linebreak[4] 
[SubtractEquations(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subtractEquations(R)
\mbox{''}
]
\linebreak[4] 
[NeqAddition(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}neqAddition(R)
\mbox{''}
]
\linebreak[4] 
[PositiveToRight(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Less)(R)
\mbox{''}
]
\linebreak[4] 
[PositiveToRight(Less)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]
\linebreak[4] 
[LeqNeqLess(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqNeqLess(R)
\mbox{''}
]
\linebreak[4] 
[SubLeqLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subLeqLeft(R)
\mbox{''}
]
\linebreak[4] 
[ToLeq(Advanced)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}toLeq(Advanced)(R)
\mbox{''}
]
\linebreak[4] 
[LeqLessTransitivity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqLessTransitivity(R)
\mbox{''}
]
\linebreak[4] 
[NegativeToLeft(Eq)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToLeft(Eq)(R)
\mbox{''}
]
\linebreak[4] 
[NegativeToRight(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToRight(Less)(R)
\mbox{''}
]
\linebreak[4] 
[!!==Symmetry
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}!!
{=}
{=}Symmetry
\mbox{''}
]
\linebreak[4] 
[SwitchTerms(x<=y-z)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}switchTerms(x
{<}
{=}y
\mbox{-}z)
\mbox{''}
]
\linebreak[4] 
[Plus0Left(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}plus0Left(R)
\mbox{''}
]
\linebreak[4] 
[PositiveToRight(Eq)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Eq)(R)
\mbox{''}
]
\linebreak[4] 
[EqAdditionLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eqAdditionLeft(R)
\mbox{''}
]
\linebreak[4] 
[Three2twoTerms(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}three2twoTerms(R)
\mbox{''}
]
\linebreak[4] 
[To!!==
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}to!!
{=}
{=}
\mbox{''}
]
\linebreak[4] 
[PositiveToRight(Less)(1 term)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]
\linebreak[4] 
[(A~)to(~E)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(A
\char126 )to(
\char126 E)
\mbox{''}
]
\linebreak[4] 
[NegativeToRight(Eq)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToRight(Eq)(R)
\mbox{''}
]
\linebreak[4] 
[NegativeToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]
\linebreak[4] 
[DoubleMinus(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}doubleMinus(R)
\mbox{''}
]
\linebreak[4] 
[UniqueNegative(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}uniqueNegative(R)
\mbox{''}
]
\linebreak[4] 
[SubtractEquationsLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subtractEquationsLeft(R)
\mbox{''}
]
\linebreak[4] 
[EqNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eqNegated(R)
\mbox{''}
]
\linebreak[4] 
[NeqNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}neqNegated(R)
\mbox{''}
]
\linebreak[4] 
[-0=0(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}0
{=}0(R)
\mbox{''}
]
\linebreak[4] 
[NegativeNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeNegated(R)
\mbox{''}
]
\linebreak[4] 
[FromLeqGeq(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}leqGeq(R)
\mbox{''}
]
\linebreak[4] 
[0<=|x|(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}
{=}|x|(R)
\mbox{''}
]
\linebreak[4] 
[PositiveNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveNegated(R)
\mbox{''}
]
\linebreak[4] 
[AddEquations(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}addEquations(R)
\mbox{''}
]
\linebreak[4] 
[Times(-1)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}times(
\mbox{-}1)(R)
\mbox{''}
]
\linebreak[4] 
[Times(-1)Left(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}times(
\mbox{-}1)Left(R)
\mbox{''}
]
\linebreak[4] 
[-x-y=-(x+y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}x
\mbox{-}y
{=}
\mbox{-}(x
{+}y)(R)
\mbox{''}
]
\linebreak[4] 
[LessTotality(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessTotality(R)
\mbox{''}
]
\linebreak[4] 
[SameNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameNumerical(R)
\mbox{''}
]
\linebreak[4] 
[MinusNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}minusNegated(R)
\mbox{''}
]
\linebreak[4] 
[PositiveNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveNumerical(R)
\mbox{''}
]
\linebreak[4] 
[SignNumerical(+)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}signNumerical(
{+})(R)
\mbox{''}
]
\linebreak[4] 
[NonnegativeNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonnegativeNumerical(R)
\mbox{''}
]
\linebreak[4] 
[NegativeNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeNumerical(R)
\mbox{''}
]
\linebreak[4] 
[LeqNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqNegated(R)
\mbox{''}
]
\linebreak[4] 
[LessNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNegated(R)
\mbox{''}
]
\linebreak[4] 
[SubLeqRight(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subLeqRight(R)
\mbox{''}
]
\linebreak[4] 
[FromLess(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}fromLess(R)
\mbox{''}
]
\linebreak[4] 
[DistributionOut(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}distributionOut(R)
\mbox{''}
]
\linebreak[4] 
[x*0+x=x(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{*}0
{+}x
{=}x(R)
\mbox{''}
]
\linebreak[4] 
[x*0=0(R)(fff)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{*}0
{=}0(R)fff
\mbox{''}
]
\linebreak[4] 
[SignNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}signNumerical(R)
\mbox{''}
]
\linebreak[4] 
[NumericalDifference(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}numericalDifference(R)
\mbox{''}
]
\linebreak[4] 
[x<=|x|(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{<}
{=}|x|(R)
\mbox{''}
]
\linebreak[4] 
[USlimitIsUpperBound(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USlimitIsUpperBound
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]
\linebreak[4] 
[USlimitIsUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USlimitIsUpperBound
\mbox{''}
]
\linebreak[4] 
[(-1)*(-1)+(-1)*1=0(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(
\mbox{-}1)
{*}(
\mbox{-}1)
{+}(
\mbox{-}1)
{*}1
{=}0(R)
\mbox{''}
]
\linebreak[4] 
[(-1)*(-1)=1(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(
\mbox{-}1)
{*}(
\mbox{-}1)
{=}1(R)
\mbox{''}
]
\linebreak[4] 
[0<1Helper(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}1Helper(R)
\mbox{''}
]
\linebreak[4] 
[0<1(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}1(R)
\mbox{''}
]
\linebreak[4] 
[ExpZero(Exact)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expZero
\linebreak [0]\ \hskip0em plus2.0em{}exact(R)
\mbox{''}
]
\linebreak[4] 
[PositiveBase(R)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase(R)
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[Three2twoFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}three2twoFactors(R)
\mbox{''}
]
\linebreak[4] 
[x=x*y*(1/y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{=}x
{*}y
{*}(1/y)(R)
\mbox{''}
]
\linebreak[4] 
[NeqMultiplication(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}neqMultiplication(R)
\mbox{''}
]
\linebreak[4] 
[LessTransitivity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessTransitivity(R)
\mbox{''}
]
\linebreak[4] 
[0<2(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}2(R)
\mbox{''}
]
\linebreak[4] 
[SameExp(R)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp(R)
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[SameExp(R)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp(R)
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[SameExp(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp(R)
\mbox{''}
]
\linebreak[4] 
[SubNeqLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subNeqLeft(R)
\mbox{''}
]
\linebreak[4] 
[SubNeqRight(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subNeqRight(R)
\mbox{''}
]
\linebreak[4] 
[NonzeroFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonzeroFactors(R)
\mbox{''}
]
\linebreak[4] 
[NonnegativeFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonnegativeFactors(R)
\mbox{''}
]
\linebreak[4] 
[PositiveFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveFactors(R)
\mbox{''}
]
\linebreak[4] 
[LessDivision(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessDivision(R)
\mbox{''}
]
\linebreak[4] 
[0<1/2(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}1/2(R)
\mbox{''}
]
\linebreak[4] 
[PositiveToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]
\linebreak[4] 
[Exp(+1)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}exp(
{+}1)(R)
\mbox{''}
]
\linebreak[4] 
[PositiveBase(R)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase(R)
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[PositiveBase(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase(R)
\mbox{''}
]
\linebreak[4] 
[-x*y=-(x*y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}x
{*}y
{=}
\mbox{-}(x
{*}y)(R)
\mbox{''}
]
\linebreak[4] 
[PositiveToLeft(Eq)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToLeft(Eq)(R)
\mbox{''}
]
\linebreak[4] 
[Times1Left(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}times1Left(R)
\mbox{''}
]
\linebreak[4] 
[x+x=2*x(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{+}x
{=}2
{*}x(R)
\mbox{''}
]
\linebreak[4] 
[(1/2)x+(1/2)x=x(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(1/2)x
{+}(1/2)x
{=}x(R)
\mbox{''}
]
\linebreak[4] 
[DistributionOut(Minus)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}distributionOut(Minus)(R)
\mbox{''}
]
\linebreak[4] 
[(1/2)(x+y)-x=(1/2)(y-x)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(1/2)(x
{+}y)
\mbox{-}x
{=}(1/2)(y
\mbox{-}x)(R)
\mbox{''}
]
\linebreak[4] 
[IntervalSize(R)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intervalSize(R)
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[LessMultiplicationLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessMultiplicationLeft(R)
\mbox{''}
]
\linebreak[4] 
[NegativeToLeft(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToLeft(Less)(R)
\mbox{''}
]
\linebreak[4] 
[NegativeToLeft(Less)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToLeft(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]
\linebreak[4] 
[y-(1/2)(x+y)=(1/2)(y-x)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}y
\mbox{-}(1/2)(x
{+}y)
{=}(1/2)(y
\mbox{-}x)(R)
\mbox{''}
]
\linebreak[4] 
[IntervalSize(R)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intervalSize(R)
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[IntervalSize(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intervalSize(R)
\mbox{''}
]
\linebreak[4] 
[XSlessUS(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}XSlessUS(R)
\mbox{''}
]
\linebreak[4] 
[USdecreasing(+1)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing(
{+}1)(R)
\mbox{''}
]
\linebreak[4] 
[ExpUnbounded(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expUnbounded
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[ExpUnbounded(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expUnbounded
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[ExpUnbounded
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expUnbounded
\mbox{''}
]
\linebreak[4] 
[1<=x+1(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}1
{<}
{=}x
{+}1(N)
\mbox{''}
]
\linebreak[4] 
[ExpNonzero(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expNonzero
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[ExpNonzero(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expNonzero
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[ExpNonzero
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expNonzero
\mbox{''}
]
\linebreak[4] 
[ExpNonzero(2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expNonzero(2)
\mbox{''}
]
\linebreak[4] 
[HalfBase(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}halfBase
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[HalfBase(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}halfBase
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[MultiplyEquations(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}multiplyEquations(R)
\mbox{''}
]
\linebreak[4] 
[NonreciprocalToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonreciprocalToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]
\linebreak[4] 
[PositiveNonzero(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveNonzero(R)
\mbox{''}
]
\linebreak[4] 
[NonzeroProduct(2)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonzeroProduct(2)(R)
\mbox{''}
]
\linebreak[4] 
[HalfBase
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}halfBase
\mbox{''}
]
\linebreak[4] 
[Three2threeFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}three2threeFactors(R)
\mbox{''}
]
\linebreak[4] 
[x*y=zBackwards(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{*}y
{=}zBackwards(R)
\mbox{''}
]
\linebreak[4] 
[PositiveInverted(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveInverted(R)
\mbox{''}
]
\linebreak[4] 
[ReciprocalToRight(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}reciprocalToRight(Less)(R)
\mbox{''}
]
\linebreak[4] 
[ReciprocalToRight(Less)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}reciprocalToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]
\linebreak[4] 
[NonreciprocalToLeft(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonreciprocalToLeft(Less)(R)
\mbox{''}
]
\linebreak[4] 
[1<x*y(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}1
{<}x
{*}y(R)
\mbox{''}
]
\linebreak[4] 
[SwitchFactors(1/x<y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}switchFactors(1/x
{<}y)(R)
\mbox{''}
]
\linebreak[4] 
[SmallHalving
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}smallHalving
\mbox{''}
]
\linebreak[4] 
[IntervalSize(anyPositive)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intervalSize(anyPositive)
\mbox{''}
]
\linebreak[4] 
[USdecreasing(+n)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing(
{+}n)
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]
\linebreak[4] 
[USdecreasing(+n)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing(
{+}n)
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]
\linebreak[4] 
[USdecreasing(+n)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing(
{+}n)
\mbox{''}
]
\linebreak[4] 
[USdecreasing
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing
\mbox{''}
]
\linebreak[4] 
[LeqAdditionLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqAdditionLeft(R)
\mbox{''}
]
\linebreak[4] 
[ToNotLess(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}toNotLess(R)
\mbox{''}
]
\linebreak[4] 
[LimitOfUSIsLeq
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}limitOfUSIsLeq
\mbox{''}
]
\linebreak[4] 
[SubtractEquations(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subtractEquations(Less)(R)
\mbox{''}
]
\linebreak[4] 
[SubtractEquationsLeft(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subtractEquationsLeft(Less)(R)
\mbox{''}
]
\linebreak[4] 
[LessNegated(Negative)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNegated(Negative)(R)
\mbox{''}
]
\linebreak[4] 
[FromNegatedAnd(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}and
\linebreak [0]\ \hskip0em plus2.0em{}(imply)
\mbox{''}
]
\linebreak[4] 
[RemoveDoubleNeg(Consequent)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}remove
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}neg
\linebreak [0]\ \hskip0em plus2.0em{}(consequent)
\mbox{''}
]
\linebreak[4] 
[FromNotUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}fromNotUpperBound
\mbox{''}
]
\linebreak[4] 
[LeqNUB
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqNUB
\mbox{''}
]
\linebreak[4] 
[USlimitIsLeastUpperBound(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USlimitIsLeastUpperBound
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]
\linebreak[4] 
[USlimitIsLeastUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USlimitIsLeastUpperBound
\mbox{''}
]
\linebreak[4] 
[ExistMP3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}exist
\linebreak [0]\ \hskip0em plus2.0em{}mp3
\mbox{''}
]
\linebreak[4] 
[GreaterPositive(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}greaterPositive(N)
\mbox{''}
]
\linebreak[4] 
[ysFClose(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}ysFClose
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]
\linebreak[4] 
[ysFClose
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}ysFClose
\mbox{''}
]
\linebreak[4] 
[ysFCauchy(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}ysFCauchy
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]
\linebreak[4] 
[ysFCauchy
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}ysFCauchy
\mbox{''}
]
\linebreak[4] 
[from<<==
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
{<}
{<}
{=}
{=}
\mbox{''}
]
\linebreak[4] 
[to<<==
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}to
{<}
{<}
{=}
{=}
\mbox{''}
]
\linebreak[4] 
[NonnegativeNumerical(F)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonnegativeNumerical(F)
\mbox{''}
]
\linebreak[4] 
[NegativeNumerical(F)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeNumerical(F)
\mbox{''}
]
\linebreak[4] 
[
\mbox{tester1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester1
\mbox{''}
]
\linebreak[4] 
[
\mbox{tester2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester2
\mbox{''}
]
\linebreak[4] 
[
\mbox{tester3}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester3
\mbox{''}
]
\linebreak[4] 
[
\mbox{tester4}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester4
\mbox{''}
]
\linebreak[4] 
[
\mbox{tester5}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester5
\mbox{''}
]
\linebreak[4] 
[
\mbox{tester6}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester6
\mbox{''}
]
\linebreak[4] 
[sup2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sup2
\mbox{''}
]
\linebreak[4] 
)^{\mathbf {p}}$
\end{flushleft}

\newpage

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

\item $
[sup2
\stackrel {\mathrm {tex}}{=} 
\mbox{``}sup2
\mbox{''}
]$ 

\item $
[LeqTotality(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LeqTotality(R)
\mbox{''}
]$  
\item $
[PositiveToLeft(Eq)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToLeft(Eq)
\mbox{''}
]$ 
\item $
[ExpZero(Exact) 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpZero(Exact)
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$  
\item $
[(+1)IsPositive(N)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(
{+}1)IsPositive(N)
\mbox{''}
]$  
\item $
[SameExp(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameExp(Base)
\mbox{''}
]$  
\item $
[SameExp(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameExp(Indu)
\mbox{''}
]$ 
\item $
[SameExp
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameExp
\mbox{''}
]$  
\item $
[Exp(+1)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Exp(
{+}1)
\mbox{''}
]$  
\item $
[DistributionOut(Minus)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}DistributionOut(Minus)
\mbox{''}
]$  
\item $
[(1/2)(x+y)-x=(1/2)(y-x)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(1/2)(x
{+}y)
\mbox{-}x
{=}(1/2)(y
\mbox{-}x)
\mbox{''}
]$  
\item $
[y-(1/2)(x+y)=(1/2)(y-x)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}y
\mbox{-}(1/2)(x
{+}y)
{=}(1/2)(y
\mbox{-}x)
\mbox{''}
]$  
\item $
[PositiveBase(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveBase(Base)
\mbox{''}
]$  
\item $
[PositiveBase(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveBase(Indu)
\mbox{''}
]$  
\item $
[PositiveBase
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveBase
\mbox{''}
]$  
\item $
[PositiveToRight(Eq)(1 term)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$  
\item $
[BSzero(Exact)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}BSzero(Exact)
\mbox{''}
]$  
\item $
[SameBS(2)(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameBS(2)(Base)
\mbox{''}
]$  
\item $
[SameBS(2)(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameBS(2)(Indu)
\mbox{''}
]$  
\item $
[SameBS(2)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameBS(2)
\mbox{''}
]$  
\item $
[NegativeToLeft(Less)(1 term)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeToLeft(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$ 

\item $
[BS(+1)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}BS(
{+}1)
\mbox{''}
]$  
\item $
[BSbound(Exact)(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}BSbound(Exact)(Base)
\mbox{''}
]$  
\item $
[BSbound(Exact)(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}BSbound(Exact)(Indu)
\mbox{''}
]$ 
\item $
[BSbound(Exact)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}BSbound(Exact)
\mbox{''}
]$ 
\item $
[BSbound
\stackrel {\mathrm {tex}}{=} 
\mbox{``}BSbound
\mbox{''}
]$ 
\item $
[SameSeries(NumDiff)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameSeries(NumDiff)
\mbox{''}
]$  
\item $
[UStelescope(Zero)(Exact)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}UStelescope(Zero)(Exact)
\mbox{''}
]$  
\item $
[SameTelescope(2)(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameTelescope(2)(Base)
\mbox{''}
]$  
\item $
[SameTelescope(2)(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameTelescope(2)(Indu)
\mbox{''}
]$ 
\item $
[SameTelescope(2)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameTelescope(2)
\mbox{''}
]$ 
\item $
[TelescopeNumerical(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}TelescopeNumerical(Base)
\mbox{''}
]$  
\item $
[TelescopeNumerical(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}TelescopeNumerical(Indu)
\mbox{''}
]$ 
\item $
[TelescopeNumerical
\stackrel {\mathrm {tex}}{=} 
\mbox{``}TelescopeNumerical
\mbox{''}
]$ 
\item $
[ToNegatedDoubleImply
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ToNegatedDoubleImply
\mbox{''}
]$  
\item $
[EqAdditionLeft(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqAdditionLeft(R)
\mbox{''}
]$  
\item $
[x=x+(y-y)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}x
{=}x
{+}(y
\mbox{-}y)(R)
\mbox{''}
]$  
\item $
[x=x+y-y(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}x
{=}x
{+}y
\mbox{-}y(R)
\mbox{''}
]$  
\item $
[Three2twoTerms(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Three2twoTerms(R)
\mbox{''}
]$  
\item $
[PositiveToRight(Less)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToRight(Less)(R)
\mbox{''}
]$  
\item $
[Three2threeTerms(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Three2threeTerms(R)
\mbox{''}
]$  
\item $
[Plus0Left(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Plus0Left(R)
\mbox{''}
]$  
\item $
[PositiveToRight(Eq)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToRight(Eq)(R)
\mbox{''}
]$  
\item $
[SubtractEquations(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubtractEquations(R)
\mbox{''}
]$  
\item $
[NeqAddition(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NeqAddition(R)
\mbox{''}
]$  
\item $
[PositiveToRight(Less)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToRight(Less)(R)
\mbox{''}
]$  
\item $
[PositiveToRight(Less)(1 term)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$ 
\item $
[To!!==
\stackrel {\mathrm {tex}}{=} 
\mbox{``}To!!
{=}
{=}
\mbox{''}
]$  
\item $
[SwitchTerms(x<=y-z)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SwitchTerms(x
{<}
{=}y
\mbox{-}z)
\mbox{''}
]$  
\item $
[(A)to(~E~)(Imply)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(A)to(
\char126 E
\char126 )(Imply)
\mbox{''}
]$  
\item $
[(E)to(~A~)(Imply)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(E)to(
\char126 A
\char126 )(Imply)
\mbox{''}
]$  
\item $
[(E~)to(~A)(Imply)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(E
\char126 )to(
\char126 A)(Imply)
\mbox{''}
]$  
\item $
[AddNegatedAll
\stackrel {\mathrm {tex}}{=} 
\mbox{``}AddNegatedAll
\mbox{''}
]$  
\item $
[ToNegatedAEA 
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ToNegatedAEA
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$  
\item $
[LessNeq(F)(Helper)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessNeq(F)(Helper)
\mbox{''}
]$  
\item $
[LessNeq(F)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessNeq(F)
\mbox{''}
]$  
\item $
[LessNeq(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessNeq(R)
\mbox{''}
]$  
\item $
[PositiveToRight(Less)(1 term)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$  
\item $
[(A~)to(~E)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(A
\char126 )to(
\char126 E)
\mbox{''}
]$  
\item $
[ToLeq(Advanced)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ToLeq(Advanced)(R)
\mbox{''}
]$  
\item $
[LeqNeqLess(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LeqNeqLess(R)
\mbox{''}
]$  
\item $
[SubLeqLeft(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubLeqLeft(R)
\mbox{''}
]$  
\item $
[LeqLessTransitivity(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LeqLessTransitivity(R)
\mbox{''}
]$  
\item $
[NegativeToLeft(Eq)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeToLeft(Eq)(R)
\mbox{''}
]$  
\item $
[NegativeToRight(Less)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeToRight(Less)(R)
\mbox{''}
]$  
\item $
[!!==Symmetry
\stackrel {\mathrm {tex}}{=} 
\mbox{``}!!
{=}
{=}Symmetry
\mbox{''}
]$  
\item $
[NegativeToRight(Eq)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeToRight(Eq)(R)
\mbox{''}
]$  
\item $
[NegativeToRight(Eq)(1 term)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$  
\item $
[DoubleMinus(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}DoubleMinus(R)
\mbox{''}
]$  
\item $
[UniqueNegative(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}UniqueNegative(R)
\mbox{''}
]$  
\item $
[SubtractEquationsLeft(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubtractEquationsLeft(R)
\mbox{''}
]$  
\item $
[EqNegated(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}EqNegated(R)
\mbox{''}
]$  
\item $
[NeqNegated(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NeqNegated(R)
\mbox{''}
]$  
\item $
[SubLeqRight(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubLeqRight(R)
\mbox{''}
]$  
\item $
[LeqNegated(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LeqNegated(R)
\mbox{''}
]$  
\item $
[LessNegated(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessNegated(R)
\mbox{''}
]$  
\item $
[-0=0(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox{-}0
{=}0(R)
\mbox{''}
]$  
\item $
[NegativeNegated(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeNegated(R)
\mbox{''}
]$  
\item $
[FromLeqGeq(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}FromLeqGeq(R)
\mbox{''}
]$  
\item $
[FromLess(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}FromLess(R)
\mbox{''}
]$  
\item $
[NonnegativeNumerical(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NonnegativeNumerical(R)
\mbox{''}
]$  
\item $
[NegativeNumerical(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeNumerical(R)
\mbox{''}
]$  
\item $
[0<=|x|(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}0
{<}
{=}|x|(R)
\mbox{''}
]$  
\item $
[PositiveNegated(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveNegated(R)
\mbox{''}
]$  
\item $
[AddEquations(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}AddEquations(R)
\mbox{''}
]$  
\item $
[DistributionOut(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}DistributionOut(R)
\mbox{''}
]$  
\item $
[x*0+x=x(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}x
{*}0
{+}x
{=}x(R)
\mbox{''}
]$

\item $
[x*0=0(R)(fff)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}x
{*}0
{=}0(R)(fff)
\mbox{''}
]$

\item $
[Times(-1)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Times(
\mbox{-}1)(R)
\mbox{''}
]$  
\item $
[Times(-1)Left(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Times(
\mbox{-}1)Left(R)
\mbox{''}
]$  
\item $
[-x-y=-(x+y)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox{-}x
\mbox{-}y
{=}
\mbox{-}(x
{+}y)(R)
\mbox{''}
]$  
\item $
[LessTotality(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessTotality(R)
\mbox{''}
]$  
\item $
[SameNumerical(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameNumerical(R)
\mbox{''}
]$  
\item $
[MinusNegated(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MinusNegated(R)
\mbox{''}
]$  
\item $
[PositiveNumerical(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveNumerical(R)
\mbox{''}
]$  
\item $
[SignNumerical(+)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SignNumerical(
{+})(R)
\mbox{''}
]$  
\item $
[SignNumerical(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SignNumerical(R)
\mbox{''}
]$  
\item $
[NumericalDifference(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NumericalDifference(R)
\mbox{''}
]$  
\item $
[x<=|x|(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}x
{<}
{=}|x|(R)
\mbox{''}
]$  
\item $
[USlimitIsUpperBound(Helper)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USlimitIsUpperBound(Helper)
\mbox{''}
]$  
\item $
[USlimitIsUpperBound
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USlimitIsUpperBound
\mbox{''}
]$ 
\item $
[(-1)*(-1)+(-1)*1=0(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(
\mbox{-}1)
{*}(
\mbox{-}1)
{+}(
\mbox{-}1)
{*}1
{=}0(R)
\mbox{''}
]$ 

\item $
[(-1)*(-1)=1(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(
\mbox{-}1)
{*}(
\mbox{-}1)
{=}1(R)
\mbox{''}
]$ 

\item $
[0<1Helper(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}0
{<}1Helper(R)
\mbox{''}
]$  
\item $
[0<1(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}0
{<}1(R)
\mbox{''}
]$  
\item $
[ExpZero(Exact)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpZero(Exact)(R)
\mbox{''}
]$  
\item $
[PositiveBase(R)(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveBase(R)(Base)
\mbox{''}
]$  
\item $
[Three2twoFactors(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Three2twoFactors(R)
\mbox{''}
]$  
\item $
[x=x*y*(1/y)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}x
{=}x
{*}y
{*}(1/y)(R)
\mbox{''}
]$ 

\item $
[NeqMultiplication(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NeqMultiplication(R)
\mbox{''}
]$  
\item $
[LessTransitivity(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessTransitivity(R)
\mbox{''}
]$ 
\item $
[0<2(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}0
{<}2(R)
\mbox{''}
]$  
\item $
[SameExp(R)(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameExp(R)(Base)
\mbox{''}
]$  
\item $
[SameExp(R)(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameExp(R)(Indu)
\mbox{''}
]$  
\item $
[SameExp(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SameExp(R)
\mbox{''}
]$ 
\item $
[SubNeqLeft(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubNeqLeft(R)
\mbox{''}
]$  
\item $
[SubNeqRight(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubNeqRight(R)
\mbox{''}
]$  
\item $
[NonzeroFactors(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NonzeroFactors(R)
\mbox{''}
]$  
\item $
[NonnegativeFactors(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NonnegativeFactors(R)
\mbox{''}
]$  
\item $
[PositiveFactors(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveFactors(R)
\mbox{''}
]$  
\item $
[LessDivision(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessDivision(R)
\mbox{''}
]$  
\item $
[0<1/2(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}0
{<}1/2(R)
\mbox{''}
]$  
\item $
[PositiveToRight(Eq)(1 term)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$  
\item $
[Exp(+1)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Exp(
{+}1)(R)
\mbox{''}
]$  
\item $
[PositiveBase(R)(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveBase(R)(Indu)
\mbox{''}
]$  
\item $
[PositiveBase(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveBase(R)
\mbox{''}
]$ 
\item $
[-x*y=-(x*y)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\mbox{-}x
{*}y
{=}
\mbox{-}(x
{*}y)(R)
\mbox{''}
]$ 

\item $
[PositiveToLeft(Eq)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveToLeft(Eq)(R)
\mbox{''}
]$  
\item $
[Times1Left(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Times1Left(R)
\mbox{''}
]$  
\item $
[x+x=2*x(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}x
{+}x
{=}2
{*}x(R)
\mbox{''}
]$ 

\item $
[(1/2)x+(1/2)x=x(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(1/2)x
{+}(1/2)x
{=}x(R)
\mbox{''}
]$  
\item $
[DistributionOut(Minus)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}DistributionOut(Minus)(R)
\mbox{''}
]$  
\item $
[(1/2)(x+y)-x=(1/2)(y-x)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}(1/2)(x
{+}y)
\mbox{-}x
{=}(1/2)(y
\mbox{-}x)(R)
\mbox{''}
]$  
\item $
[IntervalSize(R)(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}IntervalSize(R)(Base)
\mbox{''}
]$ 
\item $
[LessMultiplicationLeft(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessMultiplicationLeft(R)
\mbox{''}
]$  
\item $
[NegativeToLeft(Less)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeToLeft(Less)(R)
\mbox{''}
]$  
\item $
[NegativeToLeft(Less)(1 term)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeToLeft(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$  
\item $
[y-(1/2)(x+y)=(1/2)(y-x)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}y
\mbox{-}(1/2)(x
{+}y)
{=}(1/2)(y
\mbox{-}x)(R)
\mbox{''}
]$  
\item $
[IntervalSize(R)(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}IntervalSize(R)(Indu)
\mbox{''}
]$  
\item $
[IntervalSize(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}IntervalSize(R)
\mbox{''}
]$  
\item $
[XSlessUS(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}XSlessUS(R)
\mbox{''}
]$  
\item $
[USdecreasing(+1)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USdecreasing(
{+}1)(R)
\mbox{''}
]$  
\item $
[1<=x+1(N)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}1
{<}
{=}x
{+}1(N)
\mbox{''}
]$  
\item $
[ExpUnbounded(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpUnbounded(Base)
\mbox{''}
]$  
\item $
[ExpUnbounded(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpUnbounded(Indu)
\mbox{''}
]$  
\item $
[ExpUnbounded
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpUnbounded
\mbox{''}
]$  
\item $
[NonzeroProduct(2)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NonzeroProduct(2)(R)
\mbox{''}
]$  
\item $
[PositiveNonzero(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveNonzero(R)
\mbox{''}
]$  
\item $
[NonreciprocalToRight(Eq)(1 term)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NonreciprocalToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$  
\item $
[ExpNonzero(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpNonzero(Base)
\mbox{''}
]$  
\item $
[ExpNonzero(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpNonzero(Indu)
\mbox{''}
]$  
\item $
[ExpNonzero
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpNonzero
\mbox{''}
]$ 
\item $
[MultiplyEquations(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}MultiplyEquations(R)
\mbox{''}
]$  
\item $
[ExpNonzero(2)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExpNonzero(2)
\mbox{''}
]$  
\item $
[HalfBase(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HalfBase(Base)
\mbox{''}
]$  
\item $
[HalfBase(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HalfBase(Indu)
\mbox{''}
]$  
\item $
[HalfBase
\stackrel {\mathrm {tex}}{=} 
\mbox{``}HalfBase
\mbox{''}
]$  
\item $
[Three2threeFactors(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Three2threeFactors(R)
\mbox{''}
]$ 
\item $
[x*y=zBackwards(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}x
{*}y
{=}zBackwards(R)
\mbox{''}
]$ 

\item $
[PositiveInverted(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}PositiveInverted(R)
\mbox{''}
]$  
\item $
[ReciprocalToRight(Less)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ReciprocalToRight(Less)(R)
\mbox{''}
]$  
\item $
[ReciprocalToRight(Less)(1 term)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ReciprocalToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$  
\item $
[NonreciprocalToLeft(Less)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NonreciprocalToLeft(Less)(R)
\mbox{''}
]$  
\item $
[1<x*y(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}1
{<}x
{*}y(R)
\mbox{''}
]$ 

\item $
[SwitchFactors(1/x<y)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SwitchFactors(1/x
{<}y)(R)
\mbox{''}
]$  
\item $
[SmallHalving
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SmallHalving
\mbox{''}
]$  
\item $
[IntervalSize(anyPositive)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}IntervalSize(anyPositive)
\mbox{''}
]$  
\item $
[USdecreasing(+n)(Base)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USdecreasing(
{+}n)(Base)
\mbox{''}
]$  
\item $
[USdecreasing(+n)(Indu)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USdecreasing(
{+}n)(Indu)
\mbox{''}
]$  
\item $
[USdecreasing(+n)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USdecreasing(
{+}n)
\mbox{''}
]$  
\item $
[USdecreasing
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USdecreasing
\mbox{''}
]$ 
\item $
[LeqAdditionLeft(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LeqAdditionLeft(R)
\mbox{''}
]$  
\item $
[ToNotLess(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ToNotLess(R)
\mbox{''}
]$  
\item $
[LimitOfUSIsLeq
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LimitOfUSIsLeq
\mbox{''}
]$ 
\item $
[SubtractEquations(Less)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubtractEquations(Less)(R)
\mbox{''}
]$  
\item $
[SubtractEquationsLeft(Less)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}SubtractEquationsLeft(Less)(R)
\mbox{''}
]$  
\item $
[LessNegated(Negative)(R)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LessNegated(Negative)(R)
\mbox{''}
]$  
\item $
[FromNegatedAnd(Imply)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}FromNegatedAnd(Imply)
\mbox{''}
]$  
\item $
[RemoveDoubleNeg(Consequent)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}RemoveDoubleNeg(Consequent)
\mbox{''}
]$  
\item $
[FromNotUpperBound
\stackrel {\mathrm {tex}}{=} 
\mbox{``}FromNotUpperBound
\mbox{''}
]$  
\item $
[LeqNUB
\stackrel {\mathrm {tex}}{=} 
\mbox{``}LeqNUB
\mbox{''}
]$  
\item $
[USlimitIsLeastUpperBound(Helper)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USlimitIsLeastUpperBound(Helper)
\mbox{''}
]$  
\item $
[USlimitIsLeastUpperBound
\stackrel {\mathrm {tex}}{=} 
\mbox{``}USlimitIsLeastUpperBound
\mbox{''}
]$ 
\item $
[ExistMP3
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ExistMP3
\mbox{''}
]$  
\item $
[GreaterPositive(N)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}GreaterPositive(N)
\mbox{''}
]$  
\item $
[ysFClose(Helper)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ysFClose(Helper)
\mbox{''}
]$  
\item $
[ysFClose
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ysFClose
\mbox{''}
]$  
\item $
[ysFCauchy(Helper)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ysFCauchy(Helper)
\mbox{''}
]$  
\item $
[ysFCauchy
\stackrel {\mathrm {tex}}{=} 
\mbox{``}ysFCauchy
\mbox{''}
]$ 

\item $
[from<<==
\stackrel {\mathrm {tex}}{=} 
\mbox{``}from
{<}
{<}
{=}
{=}
\mbox{''}
]$ 

\item $
[NonnegativeNumerical(F)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NonnegativeNumerical(F)
\mbox{''}
]$ 

\item $
[to<<==
\stackrel {\mathrm {tex}}{=} 
\mbox{``}to
{<}
{<}
{=}
{=}
\mbox{''}
]$ 

\item $
[NegativeNumerical(F)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}NegativeNumerical(F)
\mbox{''}
]$ 

\end{list}

\end{document}

