\documentclass [fleqn]{article}
\setlength {\overfullrule }{1mm}
\input{lgwinclude}

\usepackage{latexsym}
\usepackage{parskip}

\parindent=0pt               % Ingen indrykkning
\frenchspacing

%\usepackage{verbatim}

%\setlength{\parindent}{0em}
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% The font of each Logiweb construct is under tight control except that
% strings are typeset in whatever font is in effect at the time of
% typesetting. This is done to enhance the readability of strings in the
% TeX source generated by Logiweb. The default font for typesetting
% strings is \rm:
\everymath{\rm}

\usepackage{makeidx}
\usepackage{page}
\makeindex
\newcommand{\intro}[1]{\emph{#1}}
\newcommand{\indexintro}[1]{\index{#1}\intro{#1}}
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\hypersetup{pdfstartpage=1}
\hypersetup{pdfstartview=FitBH}
\hypersetup{pdfpagescrop={120 130 490 730}}
\hypersetup{pdftitle=Introduction to Logiweb}
\hypersetup{colorlinks=true}
\bibliographystyle{plain}

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% \afterheading suppresses indentation once, c.f. latex.ltx.
% \display{something} displays something as a displayed equation except
% that linebreaking is possible and displaymath is not turned on by default.
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\item #1\end{list}

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\item #1\end{list}

\afterheading}

\begin {document}
\title {Logic - Peano Arithmetic\\
$\quad$\\
{\large
$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)$
}
}
\author {Kasper Frederiksen, tofu@diku.dk}
\maketitle
\tableofcontents


\clearpage

\section{Introduction} 

This report proves proposition $L3.2(h)'$ in \cite{mendelson}, page 156. That is:

\display{$
[ 
S'
\mathbf{\ lemma\ } L3.2(h)'
\colon 
\dot{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
]$}

The successful proof of this lemma can be seen in section
\ref{res.lemma}.

The entire text has been verified as mathematically correct by the
LogiWeb system. The source code can be obtained from the online
version of this text:
\begin{center}
 \verb|http://www.diku.dk/hjemmesider/studerende/tofu/Logik/|
\end{center}

\subsection{About LogiWeb} 

The LogiWeb system is being developed by Klaus Grue (DIKU) and is
fully defined on the LogiWeb {\bf base page}
\footnote{This report uses the version of base page that has the time stamp {\scriptsize \bf GRD-2005-06-22-UTC-06-58-05-413682}}
written by Klaus Grue:

\begin{center}
\verb|http://www.diku.dk/~grue/logiweb/20050502/home/grue/base/latest/|
\end{center}

From a users point of view LogiWeb consists of two parts, a compiler
and a (web)server. The user will write his text using latex syntax
and whenever he wishes to express mathematics this will be written
amongst the ordinary text, but using \verb|pyk| syntax. When the
source file is compiled two things happen. The pyk part of the file is
translated into latex and given to the latex compiler along with the
rest of the text, also the mathematics expressed by the pyk is
checked for correctness. 

The LogiWeb system allows the user to reference mathematical constructs
defined on other pages. The task of managing requests for these pages
falls to the LogiWeb server. The LogiWeb server on DIKU has a
graphical front-end in the form of a webserver, that can be accessed
through the links given above.


\subsection{About the proof system} 

All proofs in this text have been completed in the axiomatic
arithmetic system called $
S'$. This
system is a Peano Arithmetic system and is fully defined on the {\bf
peano page} \footnote{This report uses the version of peano page that
has the time stamp {\scriptsize \bf GRD-2005-06-22-UTC-07-23-31-271829
}} written by Klaus Grue: \begin{center}
\verb|http://www.diku.dk/~grue/logiweb/20050502/home/grue/peano/|
\end{center}

It should be noted that the peano page redefines all the arithmetic
and logical operators from base page in special ``peano
versions''. These new operators are identical to their classical
counterparts. To avoid any confusion figure (\ref{aaa}) gives an
overview of the new operators.

\begin{figure}[h]
  \centering
\begin{tabular}{|ccc|}
\hline
$ False \Rightarrow True $ & $\to$ & $
\mathsf {F}
\mathrel{\dot{\Rightarrow}} 
\mathsf {T}$ \\
$ \mathrm{ arbitrary variable }: A, B, C,... $ & $\to$ & ${\cal A}$, ${\cal B}$, ${\cal C}$,$\ldots$\\
$ \mathrm{ specific terms }: a, b, c, ...    $ & $\to$ & $
\dot{\mathit{a}}$, $
\dot{\mathit{b}}$, $
\dot{\mathit{c}}$,$\ldots$\\
$ \forall a: a + 0 = a  $ & $\to$ & $
\dot{\forall} 
\dot{\mathit{a}}
\colon 
(
\dot{\mathit{a}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{\mathit{a}}
)$ \\
$ \neg(1 \cdot 0 = 1)  $ & $\to$ & $
\dot{\neg}\, 
(
\dot{1}
\mathop{\dot{\cdot}} 
\dot{0}
\stackrel{p}{=} 
\dot{1}
)$\\
\hline
\end{tabular}  
\caption{The new version of the operators}
  \label{aaa}
\end{figure}

Peano arithmetic is only defined on natural numbers, and the successor
operation is written like this: $
\dot{0}
'
\stackrel{p}{=} 
\dot{1}$.

This peano page also defines the axioms and rules of the $
S'$ system. These are identical to those defined
in \cite{mendelson}, chapters two and three. Below I repeat the theory
as it is defined on the peano page. Since the pyk syntax is not always
obvious I have included some comments.

System $[
S']$ is defined thus:

\display{$
[ \mathbf{Theory\ } 
S'
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
A1'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon {\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal A}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
A2'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon 
\forall {\cal C}
\colon 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} {\cal A}
\mathrel{\dot{\Rightarrow}} {\cal C}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
A3'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon 
(
\dot{\neg}\, {\cal B}
\mathrel{\dot{\Rightarrow}} 
\dot{\neg}\, {\cal A}
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\neg}\, {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal A}
)
\mathrel{\dot{\Rightarrow}} {\cal B}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
A4'
\colon 
\forall {\cal C}
\colon 
\forall {\cal A}
\colon 
\forall {\cal X}
\colon 
\forall {\cal B}
\colon 
\lceil {\cal A}
\rceil 
{\equiv}\langle 
\lceil {\cal B}
\rceil 
|
\lceil {\cal X}
\rceil 
:=
\lceil {\cal C}
\rceil 
\rangle
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\dot{\forall} {\cal X}
\colon {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal A}
]$}
This axiom allows us to remove a universal kvantor and replace the quantified 
variable with a specific term, as long as this does not cause a variable clash.
When using the axiom, you must state which term you wish to substitute into
the place of the quantified variable. The axiom is used like this:\\
$
A4'
\mathop {\char64} 
\dot{\mathit{t}}$
$\to$ 
$
\dot{\forall} 
\dot{\mathit{q}}
\colon 
\dot{\mathit{q}}
\stackrel{p}{=} 
\dot{\mathit{q}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{\mathit{t}}$

\display{$
[ 
S'
\mathbf{\ rule\ } 
A5'
\colon 
\forall {\cal X}
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon 
\dot{nonfree}(
\lceil {\cal X}
\rceil 
,
\lceil {\cal A}
\rceil 
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\dot{\forall} {\cal X}
\colon 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} {\cal A}
\mathrel{\dot{\Rightarrow}} 
\dot{\forall} {\cal X}
\colon {\cal B}
]$}
This axiom is not used in this report.

\display{$
[ 
S'
\mathbf{\ rule\ } 
MP'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon {\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
\vdash {\cal A}
\vdash {\cal B}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
Gen'
\colon 
\forall {\cal X}
\colon 
\forall {\cal A}
\colon {\cal A}
\vdash 
\dot{\forall} {\cal X}
\colon {\cal A}
]$}
This is the generalization rule, that allows us to add a universal kvantor in front
of a proof line. 

\display{$
[ 
S'
\mathbf{\ rule\ } 
S1'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon 
\forall {\cal C}
\colon {\cal A}
\stackrel{p}{=} {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal A}
\stackrel{p}{=} {\cal C}
\mathrel{\dot{\Rightarrow}} {\cal B}
\stackrel{p}{=} {\cal C}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
S2'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon {\cal A}
\stackrel{p}{=} {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal A}
'
\stackrel{p}{=} {\cal B}
'
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
S3'
\colon 
\forall {\cal A}
\colon 
{\neg }
\dot{0}
\stackrel{p}{=} {\cal A}
'
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
S4'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon {\cal A}
'
\stackrel{p}{=} {\cal B}
'
\mathrel{\dot{\Rightarrow}} {\cal A}
\stackrel{p}{=} {\cal B}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
S5'
\colon 
\forall {\cal A}
\colon {\cal A}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} {\cal A}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
S6'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon {\cal A}
\mathop{\dot{+}} {\cal B}
'
\stackrel{p}{=} 
({\cal A}
\mathop{\dot{+}} {\cal B}
)
'
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
S7'
\colon 
\forall {\cal A}
\colon {\cal A}
\mathop{\dot{\cdot}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
S8'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon {\cal A}
\mathop{\dot{\cdot}} 
({\cal B}
'
)
\stackrel{p}{=} 
({\cal A}
\mathop{\dot{\cdot}} {\cal B}
)
\mathop{\dot{+}} {\cal A}
]$}

\display{$
[ 
S'
\mathbf{\ rule\ } 
S9'
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon 
\forall {\cal C}
\colon 
\forall {\cal X}
\colon 
\newline {\cal B}
{\equiv}\langle {\cal A}
|{\cal X}
:=
\dot{0}
\rangle
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} {\cal C}
{\equiv}\langle {\cal A}
|{\cal X}
:={\cal X}
'
\rangle
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\newline {\cal B}
\mathrel{\dot{\Rightarrow}} 
\dot{\forall} {\cal X}
\colon 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
\mathrel{\dot{\Rightarrow}} 
\dot{\forall} {\cal X}
\colon {\cal A}
]$}
This is the induction axiom. What it basically says is ``base case'' $\Rightarrow$ ``induction step'' $\Rightarrow$ ``conclusion''.

\section{The Deduction Algorithm}

Mendelson introduces the {\bf Deduction Theorem} for first-order
theories in \cite{mendelson} page 74. The proof is stated in the form
of an algorithm that will turn a proof of the the lemma  
$
S'
\vdash 
({\cal B}
\vdash {\cal C}
)$, 
in to a proof of the lemma 
$
S'
\vdash {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal C}$.

While it would be possible to implement this automatic proof converter in
pyk, as a part of this text, Klaus has advised that we do not attempt
this. We are going to follow this advise. Instead we run the algorithm
by hand and type in the converted proofs.

A god description of the deduction algorithm can be found at:
\begin{center}
\verb|http://www.diku.dk/undervisning/2005s/235/deduction.html|
\end{center}


\section{Supporting Lemmas\label{support}} 

Before we start proving the lemmas in section \ref{main.lemmas}. Let
us first prove a number of lemmas that will help make the following
proofs shorter.

Pleas note that many of the lemmas in this section were included on
a when-needed basis. If, in section \ref{main.lemmas}, you notice a
situation where one of the following lemmas could have been used to
shorten the proof, then that is an ``older'' proof from before the
supporting lemma was included.

\subsection{Corollary $C1.10a$}
The lemma 
\index{C1.10a: corollary one point ten a @C1.10a: $[C1.10a]$ corollary one point ten a}%
\index{pyk: corollary one point ten a $[C1.10a]$}%
\tex{
$[C1.10a
\stackrel {\mathrm {tex}}{=} 
\mbox{``}C1.10a
\mbox{''}
]$}$[ C1.10a%
]$\footnote{$[C1.10a
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}corollary
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}point
\linebreak [0]\ \hskip0em plus2.0em{}ten
\linebreak [0]\ \hskip0em plus2.0em{}a
\mbox{''}
]$}
is taken from \cite{mendelson} page 38.

\display{$
[ 
S'
\mathbf{\ lemma\ } C1.10a
\colon 
\forall {\cal B}
\colon 
\forall {\cal C}
\colon 
\forall {\cal D}
\colon 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal C}
\vdash {\cal C}
\mathrel{\dot{\Rightarrow}} {\cal D}
\vdash {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal D}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } C1.10a
\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 B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 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 B}
\mathrel{\dot{\Rightarrow}} {\cal C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 C}
\mathrel{\dot{\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_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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal C}
\mathrel{\dot{\Rightarrow}} {\cal D}
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal C}
\mathrel{\dot{\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_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 
$:}$
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_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 B}
\mathrel{\dot{\Rightarrow}} 
({\cal C}
\mathrel{\dot{\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_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 
$:}$
A2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal C}
\mathrel{\dot{\Rightarrow}} {\cal D}
)
)
\mathrel{\dot{\Rightarrow}} 
(
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\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_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_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 B}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\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 
$:}$
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_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 B}
\mathrel{\dot{\Rightarrow}} {\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 $


\subsection{Corollary $C1.10(b)$}
The lemma 
\index{C1.10(b): corollary one point ten b @C1.10(b): $[C1.10(b)]$ corollary one point ten b}%
\index{pyk: corollary one point ten b $[C1.10(b)]$}%
\tex{
$[C1.10(b)
\stackrel {\mathrm {tex}}{=} 
\mbox{``}C1.10(b)
\mbox{''}
]$}$[ C1.10(b)%
]$\footnote{$[C1.10(b)
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}corollary
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}point
\linebreak [0]\ \hskip0em plus2.0em{}ten
\linebreak [0]\ \hskip0em plus2.0em{}b
\mbox{''}
]$}
is taken from \cite{mendelson} page 38.

\display{$
[ 
S'
\mathbf{\ lemma\ } C1.10(b)
\colon 
\forall {\cal B}
\colon 
\forall {\cal C}
\colon 
\forall {\cal D}
\colon 
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal C}
\mathrel{\dot{\Rightarrow}} {\cal D}
)
\vdash {\cal C}
\vdash {\cal B}
\mathrel{\dot{\Rightarrow}} {\cal D}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } C1.10(b)
\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 B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 C}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 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 B}
\mathrel{\dot{\Rightarrow}} 
({\cal C}
\mathrel{\dot{\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{}$}\quad
\parbox [t]{0.4\textwidth }{${\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_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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal C}
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\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_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 
$:}$
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_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 B}
\mathrel{\dot{\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 
$:}$
A2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal C}
\mathrel{\dot{\Rightarrow}} {\cal D}
)
)
\mathrel{\dot{\Rightarrow}} 
(
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\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_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_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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\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 
$:}$
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_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 B}
\mathrel{\dot{\Rightarrow}} {\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 $

\subsection{Lemma ${A1'}^*$}
The lemma 
\index{{A1'}^*: lemma prime a one star @{A1'}^*: $[{A1'}^*]$ lemma prime a one star}%
\index{pyk: lemma prime a one star $[{A1'}^*]$}%
\tex{
$[{A1'}^*
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{A1
\mbox {'}
\}
{\char94}
{*}
\mbox{''}
]$}$[ {A1'}^*%
]$\footnote{$[{A1'}^*
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}a
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}star
\mbox{''}
]$}
is used during the deduction algorithm, when one wishes to add a hypothesis, ${\cal H}$, to a
line of a proof. This will shorten the proof by one line.

\display{$
[ 
S'
\mathbf{\ lemma\ } {A1'}^*
\colon 
\forall {\cal A}
\colon 
\forall {\cal H}
\colon 
({\cal A}
\vdash {\cal H}
\mathrel{\dot{\Rightarrow}} {\cal A}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } {A1'}^*
\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}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 H}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal A}
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\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_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$
MP'
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal H}
\mathrel{\dot{\Rightarrow}} {\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 $

\subsection{Lemma ${A2'}^*$}
Use of the axiom $
A2'$ is often followed 
by a use of the modus ponens rule. If we instead use lemma 
\index{{A2'}^*: lemma prime a two star @{A2'}^*: $[{A2'}^*]$ lemma prime a two star}%
\index{pyk: lemma prime a two star $[{A2'}^*]$}%
\tex{
$[{A2'}^*
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{A2
\mbox {'}
\}
{\char94}
{*}
\mbox{''}
]$}$[ {A2'}^*%
]$\footnote{$[{A2'}^*
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}a
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}star
\mbox{''}
]$}
, then the modus ponens is not necessary.

\display{$
[ 
S'
\mathbf{\ lemma\ } {A2'}^*
\colon 
\forall {\cal H}
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
\vdash 
(
({\cal H}
\mathrel{\dot{\Rightarrow}} {\cal A}
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } {A2'}^*
\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 H}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 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 H}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\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_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 
$:}$
A2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal H}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
\mathrel{\dot{\Rightarrow}} 
(
({\cal H}
\mathrel{\dot{\Rightarrow}} {\cal A}
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} {\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 
$:}$
MP'
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal H}
\mathrel{\dot{\Rightarrow}} {\cal A}
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\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 $

\subsection{Lemma $Double Hypothesis$}
The lemma
\index{Double Hypothesis: lemma double hyp @Double Hypothesis: $[Double Hypothesis]$ lemma double hyp}%
\index{pyk: lemma double hyp $[Double Hypothesis]$}%
\tex{
$[Double Hypothesis
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Double
\linebreak [0]\ \hskip0em plus2.0em{}Hypothesis
\mbox{''}
]$}$[ Double Hypothesis%
]$\footnote{$[Double Hypothesis
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}hyp
\mbox{''}
]$}
is used during the deduction algorithm, when one wishes to add a hypothesis, ${\cal H}$, 
to a proof line of the form ${\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}$. This will shorten the proof
by one line.

\display{$
[ 
S'
\mathbf{\ lemma\ } Double Hypothesis
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon 
\forall {\cal H}
\colon 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
\vdash 
(
({\cal H}
\mathrel{\dot{\Rightarrow}} {\cal A}
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } Double Hypothesis
\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}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 H}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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}
\mathrel{\dot{\Rightarrow}} {\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_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 
$:}${A1'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\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 
$:}${A2'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} {\cal A}
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\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 $

\subsection{Lemma $L1.8$}
Lemma 
\index{L1.8: lemma one point eight @L1.8: $[L1.8]$ lemma one point eight}%
\index{pyk: lemma one point eight $[L1.8]$}%
\tex{
$[L1.8
\stackrel {\mathrm {tex}}{=} 
\mbox{``}L1.8
\mbox{''}
]$}$[ L1.8%
]$\footnote{$[L1.8
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}point
\linebreak [0]\ \hskip0em plus2.0em{}eight
\mbox{''}
]$}
from \cite{mendelson} page 36, is used to introduce the premise at the start of the deduction 
algorithm.

\display{
$
[ 
S'
\mathbf{\ lemma\ } L1.8
\colon 
\forall {\cal B}
\colon 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } L1.8
\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 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 
$:}$
A2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal B}
\mathrel{\dot{\Rightarrow}} 
(
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
\mathrel{\dot{\Rightarrow}} 
(
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\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_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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\mathrel{\dot{\Rightarrow}} 
(
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} {\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_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 B}
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\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_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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\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 
$:}$
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_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 B}
\mathrel{\dot{\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 $

\subsection{Lemma $Tautology 1$}
The lemma that I call 
\index{Tautology 1: tautology one @Tautology 1: $[Tautology 1]$ tautology one}%
\index{pyk: tautology one $[Tautology 1]$}%
\tex{
$[Tautology 1
\stackrel {\mathrm {tex}}{=} 
\mbox{``}Tautology
\linebreak [0]\ \hskip0em plus2.0em{}1
\mbox{''}
]$}$[ Tautology 1%
]$\footnote{$[Tautology 1
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}tautology
\linebreak [0]\ \hskip0em plus2.0em{}one
\mbox{''}
]$}
is an all-round useful lemma.

\display{$
[ 
S'
\mathbf{\ lemma\ } Tautology 1
\colon 
\forall {\cal A}
\colon 
\forall {\cal B}
\colon 
\forall {\cal C}
\colon 
(
({\cal A}
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
)
\vdash 
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
)
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } Tautology 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 A}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 B}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 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}
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\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_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 
$:}$
A2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal A}
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
)
\mathrel{\dot{\Rightarrow}} 
(
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\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_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_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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\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_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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} 
(
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\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 
$:}$
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_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}
\mathrel{\dot{\Rightarrow}} 
(
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\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_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 
$:}$
A2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal B}
\mathrel{\dot{\Rightarrow}} 
(
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
)
)
\mathrel{\dot{\Rightarrow}} 
(
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\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_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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal B}
)
)
\mathrel{\dot{\Rightarrow}} 
({\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\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_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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal B}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\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 
$:}$
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_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 B}
\mathrel{\dot{\Rightarrow}} 
({\cal A}
\mathrel{\dot{\Rightarrow}} {\cal C}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $

\section{The Main Proofs \label{main.lemmas}}

The purpose of this report is the proof of $L3.2(h)'$ in section \ref{res.lemma}. In order to get that far
we must first prove a number of the lemmas that precede it in
\cite{mendelson}. Of these lemma $
L3.2(a)'$ has been proved by Klaus on the peano page and lemma L3.2(e)'
is not needed. Taking into account the support lemmas proved in
section \ref{support}, this leaves parts (b), (c), (d), (f), (g) and
of course (h), still to be proven.

Of these lemmas (f), (g) and (h) are proved using induction, which
means that we will need the Deduction Algorithm.

All of these lemmas are proved in \cite{mendelson}, so the task of
this report has been to rewrite Mendelson's proofs in explicit form
and use the deduction algorithm where necessary.


\subsection{Lemma $L3.2(b)'$}

\index{L3.2(b)': lemma prime l three two b @L3.2(b)': $[L3.2(b)']$ lemma prime l three two b}%
\index{pyk: lemma prime l three two b $[L3.2(b)']$}%
\tex{
$[L3.2(b)'
\stackrel {\mathrm {tex}}{=} 
\mbox{``}L3.2(b)
\mbox {'}
\mbox{''}
]$}$[ L3.2(b)'%
]$\footnote{$[L3.2(b)'
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}b
\mbox{''}
]$}

\display{$
[ 
S'
\mathbf{\ lemma\ } L3.2(b)'
\colon 
\forall {\cal T}
\colon 
\forall {\cal R}
\colon {\cal T}
\stackrel{p}{=} {\cal R}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal T}
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } L3.2(b)'
\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 T}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal T}
\stackrel{p}{=} {\cal R}
\mathrel{\dot{\Rightarrow}} 
({\cal T}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal T}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
L3.2(a)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal T}
\stackrel{p}{=} {\cal T}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$C1.10(b)
\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 T}
\stackrel{p}{=} {\cal R}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal T}
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $



\subsection{Lemma $L3.2(c)'$}

\index{L3.2(c)': lemma prime l three two c @L3.2(c)': $[L3.2(c)']$ lemma prime l three two c}%
\index{pyk: lemma prime l three two c $[L3.2(c)']$}%
\tex{
$[L3.2(c)'
\stackrel {\mathrm {tex}}{=} 
\mbox{``}L3.2(c)
\mbox {'}
\mbox{''}
]$}$[ L3.2(c)'%
]$\footnote{$[L3.2(c)'
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}c
\mbox{''}
]$}

\display{$
[ 
S'
\mathbf{\ lemma\ } L3.2(c)'
\colon 
\forall {\cal T}
\colon 
\forall {\cal R}
\colon 
\forall {\cal S}
\colon {\cal T}
\stackrel{p}{=} {\cal R}
\mathrel{\dot{\Rightarrow}} 
({\cal R}
\stackrel{p}{=} {\cal S}
\mathrel{\dot{\Rightarrow}} {\cal T}
\stackrel{p}{=} {\cal S}
)
]$}


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } L3.2(c)'
\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 T}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} 
({\cal R}
\stackrel{p}{=} {\cal S}
\mathrel{\dot{\Rightarrow}} {\cal T}
\stackrel{p}{=} {\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(b)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal T}
\stackrel{p}{=} {\cal R}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal T}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$C1.10a
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal T}
\stackrel{p}{=} {\cal R}
\mathrel{\dot{\Rightarrow}} 
({\cal R}
\stackrel{p}{=} {\cal S}
\mathrel{\dot{\Rightarrow}} {\cal T}
\stackrel{p}{=} {\cal S}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $



\subsection{Lemma $L3.2(d)'$}

\index{L3.2(d)': lemma prime l three two d @L3.2(d)': $[L3.2(d)']$ lemma prime l three two d}%
\index{pyk: lemma prime l three two d $[L3.2(d)']$}%
\tex{
$[L3.2(d)'
\stackrel {\mathrm {tex}}{=} 
\mbox{``}L3.2(d)
\mbox {'}
\mbox{''}
]$}$[ L3.2(d)'%
]$\footnote{$[L3.2(d)'
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}d
\mbox{''}
]$}

\display{$
[ 
S'
\mathbf{\ lemma\ } L3.2(d)'
\colon 
\forall {\cal T}
\colon 
\forall {\cal R}
\colon 
\forall {\cal S}
\colon {\cal R}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} 
({\cal S}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal S}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } L3.2(d)'
\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 T}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 R}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth ][l]{$
\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 S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(c)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal R}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} 
({\cal T}
\stackrel{p}{=} {\cal S}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$Tautology 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 T}
\stackrel{p}{=} {\cal S}
\mathrel{\dot{\Rightarrow}} 
({\cal R}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(b)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal S}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} {\cal T}
\stackrel{p}{=} {\cal S}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$C1.10a
\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 S}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} 
({\cal R}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal S}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_? \else
\global \advance \lgwproofline by 1
L\ifnum \lgwproofline <10 0\fi \number \lgwproofline
\fi 
$:}$Tautology 1
\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 R}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} 
({\cal S}
\stackrel{p}{=} {\cal T}
\mathrel{\dot{\Rightarrow}} {\cal R}
\stackrel{p}{=} {\cal S}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $


\subsection{Lemma $L3.2(f)'$}
To prove 
\index{L3.2(f)': lemma prime l three two f @L3.2(f)': $[L3.2(f)']$ lemma prime l three two f}%
\index{pyk: lemma prime l three two f $[L3.2(f)']$}%
\tex{
$[L3.2(f)'
\stackrel {\mathrm {tex}}{=} 
\mbox{``}L3.2(f)
\mbox {'}
\mbox{''}
]$}$[ L3.2(f)'%
]$\footnote{$[L3.2(f)'
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}f
\mbox{''}
]$}
using induction, we will need to prove the base case
\index{{L3.2(f)'}_{BASE}: lemma prime l three two f base @{L3.2(f)'}_{BASE}: $[{L3.2(f)'}_{BASE}]$ lemma prime l three two f base}%
\index{pyk: lemma prime l three two f base $[{L3.2(f)'}_{BASE}]$}%
\tex{
$[{L3.2(f)'}_{BASE}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{L3.2(f)
\mbox {'}
\}
\_
\{BASE
\}
\mbox{''}
]$}$[ {L3.2(f)'}_{BASE}%
]$\footnote{$[{L3.2(f)'}_{BASE}
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}f
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$} (section \ref{f.base})
and the induction step
\index{{L3.2(f)'}_{HYP}: lemma prime l three two f hyp @{L3.2(f)'}_{HYP}: $[{L3.2(f)'}_{HYP}]$ lemma prime l three two f hyp}%
\index{pyk: lemma prime l three two f hyp $[{L3.2(f)'}_{HYP}]$}%
\tex{
$[{L3.2(f)'}_{HYP}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{L3.2(f)
\mbox {'}
\}
\_
\{HYP
\}
\mbox{''}
]$}$[ {L3.2(f)'}_{HYP}%
]$\footnote{$[{L3.2(f)'}_{HYP}
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}f
\linebreak [0]\ \hskip0em plus2.0em{}hyp
\mbox{''}
]$} (section \ref{f.hyp}).

\display{$
[ 
S'
\mathbf{\ lemma\ } L3.2(f)'
\colon 
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } L3.2(f)'
\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 
$:}${L3.2(f)'}_{BASE}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{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 
$:}${L3.2(f)'}_{HYP}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S9'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{0}
)
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $

\subsubsection{The base case of $L3.2(f)'$\label{f.base}}

\display{$
[ 
S'
\mathbf{\ lemma\ } {L3.2(f)'}_{BASE}
\colon 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{0}
]$}


$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } {L3.2(f)'}_{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_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 
$:}$
S5'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{0}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{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 
$:}$L3.2(b)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{0}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathrel{\dot{\Rightarrow}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{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 
$:}$
MP'
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{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 $

\subsubsection{The induction step of $L3.2(f)'$\label{f.hyp}}

\display{$
[ 
S'
\mathbf{\ lemma\ } {L3.2(f)'}_{HYP}
\colon 
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } {L3.2(f)'}_{HYP}
\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 
$:}$L1.8
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S6'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A1'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
\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{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(d)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
Hypothesize
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
\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{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
(
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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 
\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{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
'
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $


\subsection{Lemma $L3.2(g)'$}
To prove 
\index{L3.2(g)': lemma prime l three two g @L3.2(g)': $[L3.2(g)']$ lemma prime l three two g}%
\index{pyk: lemma prime l three two g $[L3.2(g)']$}%
\tex{
$[L3.2(g)'
\stackrel {\mathrm {tex}}{=} 
\mbox{``}L3.2(g)
\mbox {'}
\mbox{''}
]$}$[ L3.2(g)'%
]$\footnote{$[L3.2(g)'
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}g
\mbox{''}
]$}
using induction, we will need to prove the base case
\index{{L3.2(g)'}_{BASE}: lemma prime l three two g @{L3.2(g)'}_{BASE}: $[{L3.2(g)'}_{BASE}]$ lemma prime l three two g}%
\index{pyk: lemma prime l three two g $[{L3.2(g)'}_{BASE}]$}%
\tex{
$[{L3.2(g)'}_{BASE}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{L3.2(g)
\mbox {'}
\}
\_
\{BASE
\}
\mbox{''}
]$}$[ {L3.2(g)'}_{BASE}%
]$\footnote{$[{L3.2(g)'}_{BASE}
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}g
\mbox{''}
]$}(section \ref{g.base})
and the induction step
\index{{L3.2(g)'}_{HYP}: lemma prime l three two g hyp @{L3.2(g)'}_{HYP}: $[{L3.2(g)'}_{HYP}]$ lemma prime l three two g hyp}%
\index{pyk: lemma prime l three two g hyp $[{L3.2(g)'}_{HYP}]$}%
\tex{
$[{L3.2(g)'}_{HYP}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{L3.2(g)
\mbox {'}
\}
\_
\{HYP
\}
\mbox{''}
]$}$[ {L3.2(g)'}_{HYP}%
]$\footnote{$[{L3.2(g)'}_{HYP}
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}g
\linebreak [0]\ \hskip0em plus2.0em{}hyp
\mbox{''}
]$}(section \ref{g.hyp}).

\display{$
[ 
S'
\mathbf{\ lemma\ } L3.2(g)'
\colon 
\dot{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } L3.2(g)'
\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 
$:}${L3.2(g)'}_{BASE}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{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 
$:}${L3.2(g)'}_{HYP}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S9'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $


\subsubsection{Base case of $L3.2(g)'$\label{g.base}}

\display{$
[ 
S'
\mathbf{\ lemma\ } {L3.2(g)'}_{BASE}
\colon 
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
)
'
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } {L3.2(g)'}_{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_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 
$:}$
S5'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{\mathit{t}}
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S5'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{\mathit{t}}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
)
'
\stackrel{p}{=} 
\dot{\mathit{t}}
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
)
'
\stackrel{p}{=} 
\dot{\mathit{t}}
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(d)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
)
\stackrel{p}{=} 
\dot{\mathit{t}}
'
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
)
'
\stackrel{p}{=} 
\dot{\mathit{t}}
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
)
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{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_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_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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
)
'
\stackrel{p}{=} 
\dot{\mathit{t}}
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
)
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{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_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_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{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{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 
$:}$
Gen'
\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{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{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 $

\subsubsection{Induction step of $L3.2(g)'$\label{g.hyp}}

\display{$
[ 
S'
\mathbf{\ lemma\ } {L3.2(g)'}_{HYP}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } {L3.2(g)'}_{HYP}
\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 
$:}%
\makebox[0.4\textwidth][l]{$Local{}\gg{}$}%
\quad%
\parbox[t]{0.4\textwidth}{${\cal H}
= 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill\makebox[0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L1.8
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal H}
\mathrel{\dot{\Rightarrow}} {\cal H}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S6'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A1'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A1'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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 H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(c)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A1'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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 H}
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S6'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A1'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A1'}^*
\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 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
MP'
\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 
\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 H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
$:}$L3.2(d)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_u \else
\if \relax \csname lgwellu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellu \fi 
$:}${A1'}^*
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_v \else
\if \relax \csname lgwellv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellv \fi 
$:}${A2'}^*
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_u \else
\if \relax \csname lgwellu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellu \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_w \else
\if \relax \csname lgwellw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellw \fi 
$:}$
MP'
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_v \else
\if \relax \csname lgwellv\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellv {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellv \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 H}
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_w \else
\if \relax \csname lgwellw\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellw {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellw \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
({\cal H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
\stackrel{p}{=} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_y \else
\if \relax \csname lgwelly\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelly {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelly \fi 
$:}$
MP'
\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 
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_y \else
\if \relax \csname lgwelly\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwelly {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwelly \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
)
'
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $


\subsection{Lemma $L3.2(h)'$\label{res.lemma}}
To prove
\index{L3.2(h)': lemma prime l three two h @L3.2(h)': $[L3.2(h)']$ lemma prime l three two h}%
\index{pyk: lemma prime l three two h $[L3.2(h)']$}%
\tex{
$[L3.2(h)'
\stackrel {\mathrm {tex}}{=} 
\mbox{``}L3.2(h)
\mbox {'}
\mbox{''}
]$}$[ L3.2(h)'%
]$\footnote{$[L3.2(h)'
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}h
\mbox{''}
]$}
using induction, we will need to prove the base case
\index{{L3.2(h)'}_{BASE}: lemma prime l three two h base @{L3.2(h)'}_{BASE}: $[{L3.2(h)'}_{BASE}]$ lemma prime l three two h base}%
\index{pyk: lemma prime l three two h base $[{L3.2(h)'}_{BASE}]$}%
\tex{
$[{L3.2(h)'}_{BASE}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{L3.2(h)
\mbox {'}
\}
\_
\{BASE
\}
\mbox{''}
]$}$[ {L3.2(h)'}_{BASE}%
]$\footnote{$[{L3.2(h)'}_{BASE}
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}h
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$}(section \ref{h.base})
and the induction step
\index{{L3.2(h)'}_{HYP}: lemma prime l three two h hyp @{L3.2(h)'}_{HYP}: $[{L3.2(h)'}_{HYP}]$ lemma prime l three two h hyp}%
\index{pyk: lemma prime l three two h hyp $[{L3.2(h)'}_{HYP}]$}%
\tex{
$[{L3.2(h)'}_{HYP}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}
\{L3.2(h)
\mbox {'}
\}
\_
\{HYP
\}
\mbox{''}
]$}$[ {L3.2(h)'}_{HYP}%
]$\footnote{$[{L3.2(h)'}_{HYP}
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}h
\linebreak [0]\ \hskip0em plus2.0em{}hyp
\mbox{''}
]$}(section \ref{h.hyp}).

\display{$
[ 
S'
\mathbf{\ lemma\ } L3.2(h)'
\colon 
\dot{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } L3.2(h)'
\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 
$:}${L3.2(h)'}_{BASE}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${L3.2(h)'}_{HYP}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S9'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
)
\mathrel{\dot{\Rightarrow}} 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
\mathrel{\dot{\Rightarrow}} 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $

\subsubsection{Base case of $L3.2(h)'$\label{h.base}}

\display{$
[ 
S'
\mathbf{\ lemma\ } {L3.2(h)'}_{BASE}
\colon 
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } {L3.2(h)'}_{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_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 
$:}$
S5'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{\mathit{t}}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(f)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A4'
\mathop {\char64} 
\dot{\mathit{t}}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
(
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(c)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{0}
\stackrel{p}{=} 
\dot{0}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $

\subsubsection{Induction step of $L3.2(h)'$\label{h.hyp}}

For proving the induction step, we will need the lemma
\index{L3.2h_{SWAP}: lemma prime l three two h hyp @L3.2h_{SWAP}: $[L3.2h_{SWAP}]$ lemma prime l three two h hyp}%
\index{pyk: lemma prime l three two h hyp $[L3.2h_{SWAP}]$}%
\tex{
$[L3.2h_{SWAP}
\stackrel {\mathrm {tex}}{=} 
\mbox{``}L3.2h
\_
\{SWAP
\}
\mbox{''}
]$}$[ L3.2h_{SWAP}%
]$\footnote{$[L3.2h_{SWAP}
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}prime
\linebreak [0]\ \hskip0em plus2.0em{}l
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}h
\linebreak [0]\ \hskip0em plus2.0em{}hyp
\mbox{''}
]$}.

\display{$
[ 
S'
\mathbf{\ lemma\ } L3.2h_{SWAP}
\colon 
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\vdash 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
]$}
$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } L3.2h_{SWAP}
\colon 
\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{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A4'
\mathop {\char64} 
\dot{\mathit{q}}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{q}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{q}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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{\mathit{q}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{q}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
Gen'
\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{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{q}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{q}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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} 
\dot{\mathit{t}}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{q}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{q}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{q}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{q}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{q}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{q}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
Gen'
\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{\forall} 
\dot{\mathit{q}}
\colon 
(
\dot{\mathit{q}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{q}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A4'
\mathop {\char64} 
\dot{\mathit{r}}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{q}}
\colon 
(
\dot{\mathit{q}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{q}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_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 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $\\
$\quad$\\
Now we are ready to prove the induction step.
$\quad$\\

\display{$
[ 
S'
\mathbf{\ lemma\ } {L3.2(h)'}_{HYP}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
]$}

$
\if\relax\csname lgwprooflinep\endcsname
\def\lgwprooflinep{x}
\newcount\lgwproofline
\fi
\begingroup
\def\insideproof{x}
\lgwproofline=0 
S'
\mathbf {\ proof\ of\ } {L3.2(h)'}_{HYP}
\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 
$:}%
\makebox[0.4\textwidth][l]{$Local{}\gg{}$}%
\quad%
\parbox[t]{0.4\textwidth}{${\cal H}
= 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
$\hfill\makebox[0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L1.8
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal H}
\mathrel{\dot{\Rightarrow}} {\cal H}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S6'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A1'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
S2'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$Double Hypothesis
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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 {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal H}
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(c)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$Double Hypothesis
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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 H}
\mathrel{\dot{\Rightarrow}} 
(
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A2'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
MP'
\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_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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(d)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$Double Hypothesis
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
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_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 H}
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$L3.2(g)'
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A4'
\mathop {\char64} 
\dot{\mathit{t}}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A4'
\mathop {\char64} 
\dot{\mathit{t}}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
MP'
\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_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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
A4'
\mathop {\char64} 
\dot{\mathit{r}}
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
(
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}$
MP'
\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_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 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{t}}
'
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
$:}$L3.2h_{SWAP}
\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 {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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 
$:}${A1'}^*
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_R \else
\if \relax \csname lgwellbigr\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellbigr {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellbigr \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{${\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \fi 
$:}${A2'}^*
\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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
\stackrel{p}{=} 
(
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
'
)
\mathrel{\dot{\Rightarrow}} 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\if \relax \csname lgwprooflinep\endcsname L_u \else
\if \relax \csname lgwellu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellu \fi 
$:}$
MP'
\rhd 
\if \relax \csname lgwprooflinep\endcsname L_t \else
\if \relax \csname lgwellt\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellt {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellt \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 H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
$\hfill \makebox [0mm][l]{\quad ;}}
\newline \makebox [0.1\textwidth]{}%
\parbox [b]{0.4\textwidth }{\raggedright
\setlength {\parindent }{-0.1\textwidth }%
\makebox [0.1\textwidth ][l]{$
\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_u \else
\if \relax \csname lgwellu\endcsname
\global \advance \lgwproofline by 1
\xdef \lgwellu {L\ifnum \lgwproofline <10 0\fi \number \lgwproofline }
\fi \lgwellu \fi 
{}\gg {}$}\quad
\parbox [t]{0.4\textwidth }{$
\dot{\forall} 
\dot{\mathit{r}}
\colon 
({\cal H}
\mathrel{\dot{\Rightarrow}} 
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
'
\stackrel{p}{=} 
\dot{\mathit{r}}
'
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
$\hfill \makebox [0mm][l]{\quad \makebox[0mm]{$\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 $

\section{Conclusion}
We have now successfully proved the lemma:
\begin{center}
$
[ 
S'
\mathbf{\ lemma\ } L3.2(h)'
\colon 
\dot{\forall} 
\dot{\mathit{t}}
\colon 
\dot{\forall} 
\dot{\mathit{r}}
\colon 
(
\dot{\mathit{t}}
\mathop{\dot{+}} 
\dot{\mathit{r}}
\stackrel{p}{=} 
\dot{\mathit{r}}
\mathop{\dot{+}} 
\dot{\mathit{t}}
)
]$
\end{center}

In retrospect some of the proofs could have been made shorter/simpler,
but this does not alter the fact that the LogiWeb system has verified that 
this text is correct.

\clearpage

\appendix

\section{The name of the page}

This defines the name of the page:

\display{$
[
\mbox{logik rapport}
\stackrel {\mathrm {pyk}}{=} 
\mbox{``}logik
\linebreak [0]\ \hskip0em plus2.0em{}rapport
\mbox{''}
]$}

\section{The Bibliography}
\begin{thebibliography}{99}
\bibitem[Mendelson, 2001]{mendelson} E. Mendelson,\\
  Introduction to Mathematical Logic, fourth edition.\\
  Chapman and Hall 2001.
\end{thebibliography}

\end{document}
