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

\usepackage{latexsym}

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

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

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

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

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

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

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

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

\begin {document}

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


\raggedright

(*** MAKROER BEGYNDER ***)

$
[ph_{1}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[ph_{1}
\mathrel {\ddot {=}}
\mathsf {a}
 _{Ph} 
]
\rceil 
)
]$

$
[ph_{2}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[ph_{2}
\mathrel {\ddot {=}}
\mathsf {b}
 _{Ph} 
]
\rceil 
)
]$

$
[ph_{3}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[ph_{3}
\mathrel {\ddot {=}}
\mathsf {c}
 _{Ph} 
]
\rceil 
)
]$

$
[ph_{4}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[ph_{4}
\mathrel {\ddot {=}}
\mathsf {d}
 _{Ph} 
]
\rceil 
)
]$

$
[ph_{5}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[ph_{5}
\mathrel {\ddot {=}}
\mathsf {e}
 _{Ph} 
]
\rceil 
)
]$

$
[ph_{6}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[ph_{6}
\mathrel {\ddot {=}}
\mathsf {f}
 _{Ph} 
]
\rceil 
)
]$



$
[
\mathsf{x}
\mathrel{\dot{\wedge}} 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
\mathrel{\dot{\wedge}} 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\,(
(
\mathsf{x}
\Rightarrow \dot{\neg}\,(
\mathsf{y}
)n
)
)n
]
\rceil 
)
]$

$
[
\mathsf{x}
 \mathrel{\dot{\vee}} 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 \mathrel{\dot{\vee}} 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\,(
\mathsf{x}
)n
\Rightarrow 
\mathsf{y}
]
\rceil 
)
]$

$
[
\mathsf{x}
\mathrel{\dot{\Leftrightarrow}} 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
\mathrel{\dot{\Leftrightarrow}} 
\mathsf{y}
\mathrel {\ddot {=}}
(
\mathsf{x}
\Rightarrow 
\mathsf{y}
)
\mathrel{\dot{\wedge}} 
(
\mathsf{y}
\Rightarrow 
\mathsf{x}
)
]
\rceil 
)
]$

$
[
\mathsf{x}
 \mathrel{\neq} 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 \mathrel{\neq} 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\,(
\mathsf{x}
\!\mathrel{==}\! 
\mathsf{y}
)n
]
\rceil 
)
]$

$
[
\mathsf{x}
 \mathrel{\notin} 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 \mathrel{\notin} 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\,(
\mathsf{x}
 \mathrel{\in} 
\mathsf{y}
)n
]
\rceil 
)
]$

$
[
\mathsf{x}
 \mathrel{\subseteq} 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 \mathrel{\subseteq} 
\mathsf{y}
\mathrel {\ddot {=}}
\forall (S1ob)
\colon ((S1ob)
 \mathrel{\in} 
\mathsf{x}
\Rightarrow (S1ob)
 \mathrel{\in} 
\mathsf{y}
)
]
\rceil 
)
]$ 

$
[\{
\mathsf{x}
\}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[\{
\mathsf{x}
\}
\mathrel {\ddot {=}}\{
\mathsf{x}
,
\mathsf{x}
\}
]
\rceil 
)
]$

$
[
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}
\mathrel {\ddot {=}}\cup \{\{
\mathsf{x}
\}
,\{
\mathsf{y}
\}
\}
]
\rceil 
)
]$

$
[
\mathsf{x}
 \cap 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 \cap 
\mathsf{y}
\mathrel {\ddot {=}} \{ ph \mathrel{\in} 
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}
\mid ph_{3}
 \mathrel{\in} 
\mathsf{x}
\mathrel{\dot{\wedge}} ph_{3}
 \mathrel{\in} 
\mathsf{y}
\}
]
\rceil 
)
]$

$
[\langle 
\mathsf{x}
,
\mathsf{y}
\rangle
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[\langle 
\mathsf{x}
,
\mathsf{y}
\rangle
\mathrel {\ddot {=}}\{\{
\mathsf{x}
\}
,\{
\mathsf{x}
,
\mathsf{y}
\}
\}
]
\rceil 
)
]$


$
[
\mathsf {r}
(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf {r}
(
\mathsf{x}
,
\mathsf{y}
)
\mathrel {\ddot {=}}\langle 
\mathsf{x}
,
\mathsf{y}
\rangle
 \mathrel{\in} 
\mathsf {r}
]
\rceil 
)
]$


$
[ReflRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[ReflRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel {\ddot {=}}
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathsf {r}
(
\mathit{s}
,
\mathit{s}
)
)
]
\rceil 
)
]$


$
[SymRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[SymRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel {\ddot {=}}
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathit{t}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathsf {r}
(
\mathit{s}
,
\mathit{t}
)
\Rightarrow 
\mathsf {r}
(
\mathit{t}
,
\mathit{s}
)
)
]
\rceil 
)
]$


$
[TransRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[TransRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel {\ddot {=}}
\newline 
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
,\linebreak [0] 
\mathit{u}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathit{t}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathit{u}
 \mathrel{\in} 
\mathsf{x}
\Rightarrow 
\mathsf {r}
(
\mathit{s}
,
\mathit{t}
)
\Rightarrow 
\mathsf {r}
(
\mathit{t}
,
\mathit{u}
)
\Rightarrow 
\mathsf {r}
(
\mathit{s}
,
\mathit{u}
)
)
]
\rceil 
)
]$


\display{$
[EqRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[EqRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel {\ddot {=}}ReflRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel{\dot{\wedge}} SymRel(
\mathsf {r}
,
\mathsf{x}
)
\mathrel{\dot{\wedge}} TransRel(
\mathsf {r}
,
\mathsf{x}
)
]
\rceil 
)
]$}



$
[{\cal BS}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[{\cal BS}
\mathrel {\ddot {=}}\underline{\mathsf {bs}}
]
\rceil 
)
]$

$
[ \mathsf {OBS}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[ \mathsf {OBS}
\mathrel {\ddot {=}}\overline{\mathsf {bs}}
]
\rceil 
)
]$


\display{
$
[[
\mathsf{x}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[[
\mathsf{x}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}
\mathrel {\ddot {=}} \{ ph \mathrel{\in} \mathsf {bs}
\mid 
\mathsf {r}
(ph_{1}
,
\mathsf{x}
)
\}
]
\rceil 
)
]$
}

\display{
$
[\mathsf {bs}
 / 
\mathsf {r}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[\mathsf {bs}
 / 
\mathsf {r}
\mathrel {\ddot {=}} \{ ph \mathrel{\in} P(\mathsf {bs}
)
\mid Ex_{20}
 \mathrel{\in} \mathsf {bs}
\mathrel{\dot{\wedge}} [Ex_{20}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}
\!\mathrel{==}\! ph_{2}
\}
]
\rceil 
)
]$
}

\display{
$
[Partition(
\mathsf {p}
,\mathsf {bs}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[Partition(
\mathsf {p}
,\mathsf {bs}
)
\mathrel {\ddot {=}}
(
\forall 
\mathit{s}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf {p}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} \mathrm{\O}
)
)
\mathrel{\dot{\wedge}} 
\newline 
(
\forall 
\mathit{s}
,\linebreak [0] 
\mathit{t}
\colon 
(
\mathit{s}
 \mathrel{\in} 
\mathsf {p}
\Rightarrow 
\mathit{t}
 \mathrel{\in} 
\mathsf {p}
\Rightarrow 
\mathit{s}
 \mathrel{\neq} 
\mathit{t}
\Rightarrow 
\mathit{s}
 \cap 
\mathit{t}
\!\mathrel{==}\! \mathrm{\O}
)
)
\mathrel{\dot{\wedge}} 
\newline \cup 
\mathsf {p}
\!\mathrel{==}\! \mathsf {bs}
]
\rceil 
)
]$
}

(*** EKSISTENS-VARIABLE ***)

\display{$
[
\mathsf{x}
 ^{Ex}
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf{x}
\stackrel{r}{=} 
\lceil 
\mathsf{x}
 _{Ex}
\rceil 
]$
}

\display{
$
[Ex_{1}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[Ex_{1}
\mathrel {\ddot {=}}
\mathsf {a}
 _{Ex}
]
\rceil 
)
]$
}

$
[Ex_{2}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[Ex_{2}
\mathrel {\ddot {=}}
\mathsf {b}
 _{Ex}
]
\rceil 
)
]$

$
[Ex_{10}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[Ex_{10}
\mathrel {\ddot {=}}
\mathsf {j}
 _{Ex}
]
\rceil 
)
]$

$
[Ex_{20}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[Ex_{20}
\mathrel {\ddot {=}}
\mathsf {t}
 _{Ex}
]
\rceil 
)
]$

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

\item  $
[\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[\langle 
\mathsf {a}
{\equiv} 
\mathsf {b}
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
\mathrel {\ddot {=}}\langle 
\lceil 
\mathsf {a}
\rceil 
{\equiv}^0 
\lceil 
\mathsf {b}
\rceil 
| 
\lceil 
\mathsf{x}
\rceil 
{:==} 
\lceil 
\mathsf {t}
\rceil 
\rangle_{Ex} 
]
\rceil 
)
]$

\item  $
[\langle 
\mathsf {a}
{\equiv}^0 
\mathsf {b}
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
\stackrel{
\mathrm {val}
}{\rightarrow}
\lambda 
\mathsf {c}
.
\mathsf{x}
 ^{Ex}
\wedge \langle 
\mathsf {a}
{\equiv}^1 
\mathsf {b}
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
]$

\item  $
[\langle 
\mathsf {a}
{\equiv}^1 
\mathsf {b}
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {a}
!
\mathsf{x}
!
\mathsf {t}
!
\newline 
\mathrm {If}(
\mathsf {b}
\stackrel{r}{=} 
\lceil 
\forall_{obj} 
\mathsf {u}
\colon 
\mathsf {v}
\rceil 
,\linebreak [0]
\mathsf {F}
,\linebreak [0]
\newline 
\mathrm {If}(
\mathsf {b}
 ^{Ex}
\wedge 
\mathsf {b}
\stackrel {t}{=}
\mathsf{x}
,\linebreak [0]
\mathsf {a}
\stackrel {t}{=}
\mathsf {t}
,\linebreak [0]
\mathrm {If}(
\newline 
\mathsf {a}
\stackrel{r}{=} 
\mathsf {b}
,\linebreak [0]\langle 
\mathsf {a}
{}^t
{\equiv}^* 
\mathsf {b}
{}^t
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
,\linebreak [0]
\mathsf {F}
)
)
)
]$

\item  $
[\langle 
\mathsf {a}
{\equiv}^* 
\mathsf {b}
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
\stackrel{
\mathrm {val}
}{\rightarrow}
\mathsf {b}
!
\mathsf{x}
!
\mathsf {t}
!
\mathrm {If}(
\mathsf {a}
,\linebreak [0]
\mathsf {T}
,\linebreak [0]
\mathrm {If}(\langle 
\mathsf {a}
{}^h
{\equiv}^1 
\mathsf {b}
{}^h
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
,\linebreak [0]\langle 
\mathsf {a}
{}^t
{\equiv}^* 
\mathsf {b}
{}^t
| 
\mathsf{x}
{:==} 
\mathsf {t}
\rangle_{Ex} 
,\linebreak [0]
\mathsf {F}
)
)
]$

\end{list}

(*** AKSIOMATISK SYSTEM ***)

$
[SystemQ
\stackrel{
stmt
}{\rightarrow}
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon R(\underline{(fx)}
)
++R(\underline{(fy)}
)
 == R(\underline{(fy)}
)
++R(\underline{(fx)}
)
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon R(\underline{(fx)}
)
**R(\underline{(fy)}
)
**R(\underline{(fz)}
)
 == R(\underline{(fx)}
)
**R(\underline{(fy)}
)
**R(\underline{(fz)}
)
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(rx)}
\colon 
\forall \underline{(ry)}
\colon \underline{(rx)}
 == \underline{(ry)}
\vdash \underline{(fx)}
 \mathrel{\in} \underline{(rx)}
\vdash \underline{(fx)}
 \mathrel{\in} \underline{(ry)}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon UB(01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
, SetOfReals
)
\vdash xs
[(\underline{
\mathsf {m}}
+1
)
]
 == xs
[\underline{
\mathsf {m}}
]
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf{y}}
 <= \underline{
\mathsf{x}}
\Rightarrow \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {s}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} \{\underline{
\mathsf{x}}
,\underline{
\mathsf{y}}
\}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {s}}
\!\mathrel{==}\! \underline{
\mathsf{x}}
)n
\Rightarrow \underline{
\mathsf {s}}
\!\mathrel{==}\! \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {s}}
\!\mathrel{==}\! \underline{
\mathsf{x}}
)n
\Rightarrow \underline{
\mathsf {s}}
\!\mathrel{==}\! \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf {s}}
 \mathrel{\in} \{\underline{
\mathsf{x}}
,\underline{
\mathsf{y}}
\}
)n
)n
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \underline{
\mathsf {n}}
 = 0
\vdash BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon (\underline{
\mathsf{x}}
+0
)
 = \underline{
\mathsf{x}}
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon R(\underline{(fx)}
)
 == R(\underline{(fy)}
)
\vdash SF(\underline{(fx)}
,\underline{(fy)}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon R(\underline{(fx)}
)
 = R(\underline{(fy)}
)
\vdash R(\underline{(fx)}
)
++R(\underline{(fz)}
)
 = R(\underline{(fy)}
)
++R(\underline{(fz)}
)
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon R(\underline{(fx)}
)
++R(0f
)
 == R(\underline{(fx)}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon (\underline{
\mathsf{x}}
*1
)
 = \underline{
\mathsf{x}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{(rx)}
\colon 
\forall \underline{(ry)}
\colon \underline{(rx)}
 == \underline{(ry)}
\vdash \underline{(ry)}
 == \underline{(rx)}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \underline{
\mathsf {m}}
 = 0
\vdash  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 = 1
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon 0
 <= \underline{
\mathsf{z}}
\Rightarrow \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow (\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
 <= (\underline{
\mathsf{y}}
*\underline{
\mathsf{z}}
)
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon R(\underline{(fx)}
)
**R(
\mbox{1f}
)
 == R(\underline{(fx)}
)
\mathrel {\oplus} \dot{\neg}\,(0
 = 1
)n
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon Nat(\underline{
\mathsf {m}}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 0
 <= \underline{
\mathsf {m}}
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(\underline{
\mathsf{x}}
\!\mathrel{==}\! \underline{
\mathsf{y}}
\Rightarrow 
\forall_{obj} \overline{
\mathsf {s}}
\colon \dot{\neg}\,(\overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {s}}
\colon \dot{\neg}\,(\overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
)n
)n
\Rightarrow \underline{
\mathsf{x}}
\!\mathrel{==}\! \underline{
\mathsf{y}}
)n
)n
\mathrel {\oplus} 
\forall \underline{(rx)}
\colon 
\forall \underline{(ry)}
\colon 
\forall \underline{(rz)}
\colon \underline{(rx)}
 == \underline{(ry)}
\vdash \underline{(ry)}
 == \underline{(rz)}
\vdash \underline{(rx)}
 == \underline{(rz)}
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon (\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
 = (\underline{
\mathsf{y}}
+\underline{
\mathsf{x}}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \underline{(fx)}
+_{f}\underline{(fy)}
[\underline{
\mathsf {m}}
]
 = (\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\mathrel {\oplus} 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \langle \underline{
\mathsf {b}}
{\equiv} \underline{
\mathsf {a}}
| \underline{(v1)}
{:==} 0
\rangle_{Me}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \langle \underline{
\mathsf {c}}
{\equiv} \underline{
\mathsf {a}}
| \underline{(v1)}
{:==} (\underline{(v1)}
+1
)
\rangle_{Me}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {b}}
\Rightarrow 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \underline{
\mathsf {n}}
 = 0
\vdash UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon R(\underline{(fx)}
)
++R(\underline{(fy)}
)
++R(\underline{(fz)}
)
 = R(\underline{(fx)}
)
++R(\underline{(fy)}
)
++R(\underline{(fz)}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon (\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
 = (\underline{
\mathsf{y}}
*\underline{
\mathsf{x}}
)
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \underline{(fx)}
 \mathrel{\in} R(\underline{(fy)}
)
\vdash SF(\underline{(fx)}
,\underline{(fy)}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\Rightarrow (\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{y}}
*\underline{
\mathsf{z}}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {a}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon UB(01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
, SetOfReals
)
\vdash us
[(\underline{
\mathsf {m}}
+1
)
]
 == 01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(\underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
)n
\Rightarrow \underline{
\mathsf{y}}
 <= \underline{
\mathsf{x}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {s}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} P(\underline{
\mathsf{x}}
)
\Rightarrow 
\forall_{obj} \overline{
\mathsf {s}}
\colon \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf {s}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {s}}
\colon \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf {s}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \underline{
\mathsf {s}}
 \mathrel{\in} P(\underline{
\mathsf{x}}
)
)n
)n
\mathrel {\oplus} us
[0
]
 == xs
[0
]
++R(
\mbox{1f}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon \underline{
\mathsf{x}}
 <= \underline{
\mathsf{x}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {s}}
\colon \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} \mathrm{\O}
)n
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{x}}
)
)
 = 0
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf{x}}
 = \underline{
\mathsf{z}}
\Rightarrow \underline{
\mathsf{y}}
 = \underline{
\mathsf{z}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf {n}}
)n
)n
)n
\vdash UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 = (|(us
[(\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+(-u1
)
)
)
)
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon R(\underline{(fx)}
+_{f}\underline{(fy)}
+_{f}\underline{(fz)}
)
 == R(\underline{(fx)}
+_{f}\underline{(fy)}
+_{f}\underline{(fz)}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(\underline{
\mathsf{x}}
 = 0
)n
\Rightarrow (\underline{
\mathsf{x}}
*rec\underline{
\mathsf{x}}
)
 = 1
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{(rx)}
\colon \underline{(rx)}
 == \underline{(rx)}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon \dot{\neg}\,(UB(01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
, SetOfReals
)
)n
\vdash us
[(\underline{
\mathsf {m}}
+1
)
]
 == us
[\underline{
\mathsf {m}}
]
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow (\underline{
\mathsf{x}}
+\underline{
\mathsf{z}}
)
 <= (\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf {p}}
 ^{Ph}
\wedge \langle \underline{
\mathsf {b}}
{\equiv} \underline{
\mathsf {a}}
| \underline{
\mathsf {p}}
{:==} \underline{
\mathsf{z}}
\rangle_{Ph} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \dot{\neg}\,(\underline{
\mathsf{z}}
 \mathrel{\in}  \{ ph \mathrel{\in} \underline{
\mathsf{x}}
\mid \underline{
\mathsf {a}}
\}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf{z}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{z}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \underline{
\mathsf{z}}
 \mathrel{\in}  \{ ph \mathrel{\in} \underline{
\mathsf{x}}
\mid \underline{
\mathsf {a}}
\}
)n
)n
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(fx)}
\colon R(\underline{(fx)}
)
++(--R(\underline{(fx)}
)
)
 == R(0f
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon (\underline{
\mathsf{x}}
*(\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
)
 = ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon Nat(\underline{
\mathsf {m}}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} Nat(\underline{
\mathsf {n}}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \dot{\neg}\,(\underline{
\mathsf {m}}
 <= (\underline{
\mathsf {n}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {m}}
 = (\underline{
\mathsf {n}}
+1
)
)n
)n
)n
\vdash \underline{
\mathsf {m}}
 <= \underline{
\mathsf {n}}
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {t}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \langle 
\lceil \underline{
\mathsf {a}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {b}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:==} 
\lceil \underline{
\mathsf {t}}
\rceil 
\rangle_{Ex} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf {m}}
)n
)n
)n
\vdash  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+(-u1
)
)
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon ((\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
+\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{x}}
+(\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
)
\mathrel {\oplus} 
\forall \underline{(v1)}
\colon 
\forall \underline{(v2)}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall_{obj} \underline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon 
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{(v1)}
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{(v2)}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{(v1)}
]
+(-u\underline{(fx)}
[\underline{(v2)}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{(v1)}
]
+(-u\underline{(fx)}
[\underline{(v2)}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \langle \underline{
\mathsf {a}}
{\equiv} \underline{
\mathsf {b}}
| \underline{(v1)}
{:==} \underline{
\mathsf{x}}
\rangle_{Me}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf {n}}
)n
)n
)n
\vdash BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+BS(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+(-u1
)
)
)
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
*\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{x}}
*(\underline{
\mathsf{y}}
*\underline{
\mathsf{z}}
)
)
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon SF(\underline{(fx)}
,\underline{(fy)}
)
\vdash R(\underline{(fx)}
)
 == R(\underline{(fy)}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\Rightarrow (\underline{
\mathsf{x}}
+\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\vdash 
\forall_{obj} \underline{
\mathsf{x}}
\colon \underline{
\mathsf {a}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {m}}
\colon \dot{\neg}\,(UB(01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
, SetOfReals
)
)n
\vdash xs
[(\underline{
\mathsf {m}}
+1
)
]
 == 01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
\mathrel {\oplus} 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf{y}}
 <= \underline{
\mathsf{z}}
\Rightarrow \underline{
\mathsf{x}}
 <= \underline{
\mathsf{z}}
\mathrel {\oplus} 
\forall \underline{
\mathsf {s}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} \cup \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} 
\mathsf {j}
 _{Ex}
\Rightarrow \dot{\neg}\,(
\mathsf {j}
 _{Ex}
 \mathrel{\in} \underline{
\mathsf{x}}
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} 
\mathsf {j}
 _{Ex}
\Rightarrow \dot{\neg}\,(
\mathsf {j}
 _{Ex}
 \mathrel{\in} \underline{
\mathsf{x}}
)n
)n
\Rightarrow \underline{
\mathsf {s}}
 \mathrel{\in} \cup \underline{
\mathsf{x}}
)n
)n
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon R(\underline{(fx)}
)
**R(\underline{(fy)}
)
 == R(\underline{(fy)}
)
**R(\underline{(fx)}
)
\mathrel {\oplus} 
\forall \underline{(fx)}
\colon 
\forall \underline{(rx)}
\colon 
\forall \underline{(ry)}
\colon \underline{(rx)}
 == \underline{(ry)}
\vdash \underline{(fx)}
 \mathrel{\in} \underline{(ry)}
\vdash \underline{(fx)}
 \mathrel{\in} \underline{(rx)}
]$

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

\item $
[MP
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
]
[MP
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[Gen
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\vdash 
\forall_{obj} \underline{
\mathsf{x}}
\colon \underline{
\mathsf {a}}
]
[Gen
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[Repetition
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {a}}
]
[Repetition
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[Neg
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash \underline{
\mathsf {b}}
]
[Neg
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[Ded
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
]
[Ded
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[ExistIntro
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {t}}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \langle 
\lceil \underline{
\mathsf {a}}
\rceil 
{\equiv}^0 
\lceil \underline{
\mathsf {b}}
\rceil 
| 
\lceil \underline{
\mathsf{x}}
\rceil 
{:==} 
\lceil \underline{
\mathsf {t}}
\rceil 
\rangle_{Ex} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
]
[ExistIntro
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[Extensionality
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(\underline{
\mathsf{x}}
\!\mathrel{==}\! \underline{
\mathsf{y}}
\Rightarrow 
\forall_{obj} \overline{
\mathsf {s}}
\colon \dot{\neg}\,(\overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {s}}
\colon \dot{\neg}\,(\overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{y}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
)n
)n
\Rightarrow \underline{
\mathsf{x}}
\!\mathrel{==}\! \underline{
\mathsf{y}}
)n
)n
]
[Extensionality
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[\O{}def
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {s}}
\colon \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} \mathrm{\O}
)n
]
[\O{}def
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[PairDef
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {s}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} \{\underline{
\mathsf{x}}
,\underline{
\mathsf{y}}
\}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {s}}
\!\mathrel{==}\! \underline{
\mathsf{x}}
)n
\Rightarrow \underline{
\mathsf {s}}
\!\mathrel{==}\! \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {s}}
\!\mathrel{==}\! \underline{
\mathsf{x}}
)n
\Rightarrow \underline{
\mathsf {s}}
\!\mathrel{==}\! \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf {s}}
 \mathrel{\in} \{\underline{
\mathsf{x}}
,\underline{
\mathsf{y}}
\}
)n
)n
]
[PairDef
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[UnionDef
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {s}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} \cup \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} 
\mathsf {j}
 _{Ex}
\Rightarrow \dot{\neg}\,(
\mathsf {j}
 _{Ex}
 \mathrel{\in} \underline{
\mathsf{x}}
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} 
\mathsf {j}
 _{Ex}
\Rightarrow \dot{\neg}\,(
\mathsf {j}
 _{Ex}
 \mathrel{\in} \underline{
\mathsf{x}}
)n
)n
\Rightarrow \underline{
\mathsf {s}}
 \mathrel{\in} \cup \underline{
\mathsf{x}}
)n
)n
]
[UnionDef
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[PowerDef
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {s}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(\underline{
\mathsf {s}}
 \mathrel{\in} P(\underline{
\mathsf{x}}
)
\Rightarrow 
\forall_{obj} \overline{
\mathsf {s}}
\colon \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf {s}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {s}}
\colon \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf {s}}
\Rightarrow \overline{
\mathsf {s}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \underline{
\mathsf {s}}
 \mathrel{\in} P(\underline{
\mathsf{x}}
)
)n
)n
]
[PowerDef
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\item $
[SeparationDef
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf {p}}
 ^{Ph}
\wedge \langle \underline{
\mathsf {b}}
{\equiv} \underline{
\mathsf {a}}
| \underline{
\mathsf {p}}
{:==} \underline{
\mathsf{z}}
\rangle_{Ph} 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \dot{\neg}\,(\underline{
\mathsf{z}}
 \mathrel{\in}  \{ ph \mathrel{\in} \underline{
\mathsf{x}}
\mid \underline{
\mathsf {a}}
\}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf{z}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{z}}
 \mathrel{\in} \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \underline{
\mathsf{z}}
 \mathrel{\in}  \{ ph \mathrel{\in} \underline{
\mathsf{x}}
\mid \underline{
\mathsf {a}}
\}
)n
)n
]
[SeparationDef
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

\end{list}

\appendix



------------- RRRRRRRRRRRRRRR  -------------

(*** import fra A.M. ***)



$
[TimesCommutativity(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon R(\underline{(fx)}
)
**R(\underline{(fy)}
)
 == R(\underline{(fy)}
)
**R(\underline{(fx)}
)
]
[TimesCommutativity(R)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


(*** aksiomer ***)

$
[leqReflexivity
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon \underline{
\mathsf{x}}
 <= \underline{
\mathsf{x}}
]
[leqReflexivity
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[leqAntisymmetryAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf{y}}
 <= \underline{
\mathsf{x}}
\Rightarrow \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
]
[leqAntisymmetryAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[leqTransitivityAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf{y}}
 <= \underline{
\mathsf{z}}
\Rightarrow \underline{
\mathsf{x}}
 <= \underline{
\mathsf{z}}
]
[leqTransitivityAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[leqTotality
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(\underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
)n
\Rightarrow \underline{
\mathsf{y}}
 <= \underline{
\mathsf{x}}
]
[leqTotality
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[leqAdditionAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow (\underline{
\mathsf{x}}
+\underline{
\mathsf{z}}
)
 <= (\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
]
[leqAdditionAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[leqMultiplicationAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon 0
 <= \underline{
\mathsf{z}}
\Rightarrow \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow (\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
 <= (\underline{
\mathsf{y}}
*\underline{
\mathsf{z}}
)
]
[leqMultiplicationAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[plusAssociativity
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon ((\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
+\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{x}}
+(\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
)
]
[plusAssociativity
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[plusCommutativity
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon (\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
 = (\underline{
\mathsf{y}}
+\underline{
\mathsf{x}}
)
]
[plusCommutativity
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[Negative
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{x}}
)
)
 = 0
]
[Negative
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[plus0
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon (\underline{
\mathsf{x}}
+0
)
 = \underline{
\mathsf{x}}
]
[plus0
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[timesAssociativity
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
*\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{x}}
*(\underline{
\mathsf{y}}
*\underline{
\mathsf{z}}
)
)
]
[timesAssociativity
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[timesCommutativity
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon (\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
 = (\underline{
\mathsf{y}}
*\underline{
\mathsf{x}}
)
]
[timesCommutativity
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[ReciprocalAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(\underline{
\mathsf{x}}
 = 0
)n
\Rightarrow (\underline{
\mathsf{x}}
*rec\underline{
\mathsf{x}}
)
 = 1
]
[ReciprocalAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[times1
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon (\underline{
\mathsf{x}}
*1
)
 = \underline{
\mathsf{x}}
]
[times1
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[Distribution
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon (\underline{
\mathsf{x}}
*(\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
)
 = ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
]
[Distribution
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[0not1
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash \dot{\neg}\,(0
 = 1
)n
]
[0not1
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[EqualityAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf{x}}
 = \underline{
\mathsf{z}}
\Rightarrow \underline{
\mathsf{y}}
 = \underline{
\mathsf{z}}
]
[EqualityAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[EqLeqAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\Rightarrow \underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
]
[EqLeqAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[EqAdditionAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\Rightarrow (\underline{
\mathsf{x}}
+\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
]
[EqAdditionAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[EqMultiplicationAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\Rightarrow (\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{y}}
*\underline{
\mathsf{z}}
)
]
[EqMultiplicationAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[A4(Axiom)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \langle \underline{
\mathsf {a}}
{\equiv} \underline{
\mathsf {b}}
| \underline{(v1)}
{:==} \underline{
\mathsf{x}}
\rangle_{Me}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
]
[A4(Axiom)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


(*** XX snydeaksiomer ***)

$
[==\!{}Reflexivity
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(rx)}
\colon \underline{(rx)}
 == \underline{(rx)}
]
[==\!{}Reflexivity
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[==\!{}Symmetry
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(rx)}
\colon 
\forall \underline{(ry)}
\colon \underline{(rx)}
 == \underline{(ry)}
\vdash \underline{(ry)}
 == \underline{(rx)}
]
[==\!{}Symmetry
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[\!{}==\!{}Transitivity
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(rx)}
\colon 
\forall \underline{(ry)}
\colon 
\forall \underline{(rz)}
\colon \underline{(rx)}
 == \underline{(ry)}
\vdash \underline{(ry)}
 == \underline{(rz)}
\vdash \underline{(rx)}
 == \underline{(rz)}
]
[\!{}==\!{}Transitivity
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


XX ikke 100procent identisk med originalen fra equivalence-relations
$
[SENC1
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(rx)}
\colon 
\forall \underline{(ry)}
\colon \underline{(rx)}
 == \underline{(ry)}
\vdash \underline{(fx)}
 \mathrel{\in} \underline{(rx)}
\vdash \underline{(fx)}
 \mathrel{\in} \underline{(ry)}
]
[SENC1
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


XX boer bevises ud fra nummer 1
$
[SENC2
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(rx)}
\colon 
\forall \underline{(ry)}
\colon \underline{(rx)}
 == \underline{(ry)}
\vdash \underline{(fx)}
 \mathrel{\in} \underline{(ry)}
\vdash \underline{(fx)}
 \mathrel{\in} \underline{(rx)}
]
[SENC2
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[PlusF
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \underline{(fx)}
+_{f}\underline{(fy)}
[\underline{
\mathsf {m}}
]
 = (\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
]
[PlusF
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[From==
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon R(\underline{(fx)}
)
 == R(\underline{(fy)}
)
\vdash SF(\underline{(fx)}
,\underline{(fy)}
)
]
[From==
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[To==
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon SF(\underline{(fx)}
,\underline{(fy)}
)
\vdash R(\underline{(fx)}
)
 == R(\underline{(fy)}
)
]
[To==
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


$
[FromInR
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \underline{(fx)}
 \mathrel{\in} R(\underline{(fy)}
)
\vdash SF(\underline{(fx)}
,\underline{(fy)}
)
]
[FromInR
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$



(*** makroer ***)

KVANTI

$
[M_{1}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[M_{1}
\mathrel {\ddot {=}}\underline{(m1)}
]
\rceil 
)
]$
$
[M_{2}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[M_{2}
\mathrel {\ddot {=}}\underline{(m2)}
]
\rceil 
)
]$
$
[N_{1} 
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[N_{1} 
\mathrel {\ddot {=}}\underline{(n1)}
]
\rceil 
)
]$
$
[N_{2} 
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[N_{2} 
\mathrel {\ddot {=}}\underline{(n2)}
]
\rceil 
)
]$
$
[N_{3} 
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[N_{3} 
\mathrel {\ddot {=}}\underline{(n3)}
]
\rceil 
)
]$
$
[\epsilon
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[\epsilon
\mathrel {\ddot {=}}\underline{(\epsilon)}
]
\rceil 
)
]$
$
[\epsilon 1
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[\epsilon 1
\mathrel {\ddot {=}}\underline{(\epsilon)_{1}}
]
\rceil 
)
]$
$
[\epsilon 2
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[\epsilon 2
\mathrel {\ddot {=}}\underline{(\epsilon 2)}
]
\rceil 
)
]$
$
[X_{1}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[X_{1}
\mathrel {\ddot {=}}\underline{(x1)}
]
\rceil 
)
]$
$
[X_{2}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[X_{2}
\mathrel {\ddot {=}}\underline{(x2)}
]
\rceil 
)
]$
$
[Y_{1}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[Y_{1}
\mathrel {\ddot {=}}\underline{(y1)}
]
\rceil 
)
]$
$
[Y_{2}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[Y_{2}
\mathrel {\ddot {=}}\underline{(y2)}
]
\rceil 
)
]$
$
[V_{1}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[V_{1}
\mathrel {\ddot {=}}\underline{(v1)}
]
\rceil 
)
]$
$
[V_{2}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[V_{2}
\mathrel {\ddot {=}}\underline{(v2)}
]
\rceil 
)
]$
$
[V_{3}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[V_{3}
\mathrel {\ddot {=}}\underline{(v3)}
]
\rceil 
)
]$ 
$
[V_{4}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[V_{4}
\mathrel {\ddot {=}}\underline{(v4)}
]
\rceil 
)
]$ 
$
[V_{2n}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[V_{2n}
\mathrel {\ddot {=}}\underline{(v2n)}
]
\rceil 
)
]$
$
[FX
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FX
\mathrel {\ddot {=}}\underline{(fx)}
]
\rceil 
)
]$
$
[FY
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FY
\mathrel {\ddot {=}}\underline{(fy)}
]
\rceil 
)
]$
$
[FZ
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FZ
\mathrel {\ddot {=}}\underline{(fz)}
]
\rceil 
)
]$
$
[FU
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FU
\mathrel {\ddot {=}}\underline{(fu)}
]
\rceil 
)
]$
$
[FV
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FV
\mathrel {\ddot {=}}\underline{(fv)}
]
\rceil 
)
]$
$
[FW
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FW
\mathrel {\ddot {=}}\underline{(fw)}
]
\rceil 
)
]$
$
[FEP
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FEP
\mathrel {\ddot {=}}\underline{(fep)}
]
\rceil 
)
]$
$
[RX
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[RX
\mathrel {\ddot {=}}\underline{(rx)}
]
\rceil 
)
]$
$
[RY
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[RY
\mathrel {\ddot {=}}\underline{(ry)}
]
\rceil 
)
]$
$
[RZ
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[RZ
\mathrel {\ddot {=}}\underline{(rz)}
]
\rceil 
)
]$
$
[RU
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[RU
\mathrel {\ddot {=}}\underline{(ru)}
]
\rceil 
)
]$
$
[(SX)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(SX)
\mathrel {\ddot {=}}\underline{(sx)}
]
\rceil 
)
]$
$
[(SX1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(SX1)
\mathrel {\ddot {=}}\underline{(sx1)}
]
\rceil 
)
]$
$
[(SY)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(SY)
\mathrel {\ddot {=}}\underline{(sy)}
]
\rceil 
)
]$
$
[(SY1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(SY1)
\mathrel {\ddot {=}}\underline{(sy1)}
]
\rceil 
)
]$
$
[(SZ)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(SZ)
\mathrel {\ddot {=}}\underline{(sz)}
]
\rceil 
)
]$
$
[(SZ1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(SZ1)
\mathrel {\ddot {=}}\underline{(sz1)}
]
\rceil 
)
]$
$
[(SU)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(SU)
\mathrel {\ddot {=}}\underline{(su)}
]
\rceil 
)
]$
$
[(SU1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(SU1)
\mathrel {\ddot {=}}\underline{(su1)}
]
\rceil 
)
]$
$
[FXS
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FXS
\mathrel {\ddot {=}}\underline{(fxs)}
]
\rceil 
)
]$
$
[FYS
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[FYS
\mathrel {\ddot {=}}\underline{(fys)}
]
\rceil 
)
]$
$
[(F1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(F1)
\mathrel {\ddot {=}}\underline{(f1)}
]
\rceil 
)
]$
$
[(F2)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(F2)
\mathrel {\ddot {=}}\underline{(f2)}
]
\rceil 
)
]$
$
[(F3)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(F3)
\mathrel {\ddot {=}}\underline{(f3)}
]
\rceil 
)
]$
$
[(F4)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(F4)
\mathrel {\ddot {=}}\underline{(f4)}
]
\rceil 
)
]$
$
[(OP1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(OP1)
\mathrel {\ddot {=}}\underline{(op1)}
]
\rceil 
)
]$
$
[(OP2)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(OP2)
\mathrel {\ddot {=}}\underline{(op2)}
]
\rceil 
)
]$
$
[(R1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(R1)
\mathrel {\ddot {=}}\underline{(r1)}
]
\rceil 
)
]$
$
[(S1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(S1)
\mathrel {\ddot {=}}\underline{(s1)}
]
\rceil 
)
]$
$
[(S2)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(S2)
\mathrel {\ddot {=}}\underline{(s2)}
]
\rceil 
)
]$

$
[(EPob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(EPob)
\mathrel {\ddot {=}}\overline{(\epsilon)}
]
\rceil 
)
]$
$
[(CRS1ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(CRS1ob)
\mathrel {\ddot {=}}\overline{(crs1)}
]
\rceil 
)
]$
$
[(F1ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(F1ob)
\mathrel {\ddot {=}}\overline{(f1)}
]
\rceil 
)
]$
$
[(F2ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(F2ob)
\mathrel {\ddot {=}}\overline{(f2)}
]
\rceil 
)
]$
$
[(F3ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(F3ob)
\mathrel {\ddot {=}}\overline{(f3)}
]
\rceil 
)
]$
$
[(F4ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(F4ob)
\mathrel {\ddot {=}}\overline{(f4)}
]
\rceil 
)
]$
$
[(N1ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(N1ob)
\mathrel {\ddot {=}}\overline{(n1)}
]
\rceil 
)
]$
$
[(N2ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(N2ob)
\mathrel {\ddot {=}}\overline{(n2)}
]
\rceil 
)
]$
$
[(OP1ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(OP1ob)
\mathrel {\ddot {=}}\overline{(op1)}
]
\rceil 
)
]$
$
[(OP2ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(OP2ob)
\mathrel {\ddot {=}}\overline{(op2)}
]
\rceil 
)
]$
$
[(R1ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(R1ob)
\mathrel {\ddot {=}}\overline{(r1)}
]
\rceil 
)
]$
$
[(S1ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(S1ob)
\mathrel {\ddot {=}}\overline{(s1)}
]
\rceil 
)
]$
$
[(S2ob)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(S2ob)
\mathrel {\ddot {=}}\overline{(s2)}
]
\rceil 
)
]$





$
[(fx)
\leq_{f}(fy)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(fx)
\leq_{f}(fy)
\mathrel {\ddot {=}}(fx)
<_{f}(fy)
 \mathrel{\dot{\vee}} SF((fx)
,(fy)
)
]
\rceil 
)
]$



$
[Ex3
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[Ex3
\mathrel {\ddot {=}}
\mathsf {c}
 _{Ex}
]
\rceil 
)
]$

$
[
\exists (v1)
\colon 
\mathsf {a}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\exists (v1)
\colon 
\mathsf {a}
\mathrel {\ddot {=}}\dot{\neg}\,(
\forall (v1)
\colon \dot{\neg}\,(
\mathsf {a}
)n
)n
]
\rceil 
)
]$

$
[
\mathsf{x}
 <<== 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 <<== 
\mathsf{y}
\mathrel {\ddot {=}}
\mathsf{x}
 << 
\mathsf{y}
 \mathrel{\dot{\vee}} 
\mathsf{x}
 == 
\mathsf{y}
]
\rceil 
)
]$

$
[(-1)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(-1)
\mathrel {\ddot {=}}(-u1
)
]
\rceil 
)
]$

$
[2
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[2
\mathrel {\ddot {=}}(1
+1
)
]
\rceil 
)
]$

$
[3
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[3
\mathrel {\ddot {=}}(2
+1
)
]
\rceil 
)
]$

$
[1/2
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[1/2
\mathrel {\ddot {=}}rec2
]
\rceil 
)
]$

$
[1/3
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[1/3
\mathrel {\ddot {=}}rec3
]
\rceil 
)
]$

$
[2/3
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[2/3
\mathrel {\ddot {=}}(2
*1/3
)
]
\rceil 
)
]$

$
[
\mathsf{x}
 < 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 < 
\mathsf{y}
\mathrel {\ddot {=}}
\mathsf{x}
 <= 
\mathsf{y}
\mathrel{\dot{\wedge}} 
\mathsf{x}
 \neq 
\mathsf{y}
]
\rceil 
)
]$

$
[
\mathsf{x}
 \neq 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 \neq 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\,(
\mathsf{x}
 = 
\mathsf{y}
)n
]
\rceil 
)
]$


$
[(
\mathsf{x}
-
\mathsf{y}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(
\mathsf{x}
-
\mathsf{y}
)
\mathrel {\ddot {=}}(
\mathsf{x}
+(-u
\mathsf{y}
)
)
]
\rceil 
)
]$

$
[00
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[00
\mathrel {\ddot {=}}R(0f
)
]
\rceil 
)
]$

$
[
\mbox{01}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mbox{01}
\mathrel {\ddot {=}}R(
\mbox{1f}
)
]
\rceil 
)
]$

$
[
\mathsf{x}
 !!== 
\mathsf{y}
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[
\mathsf{x}
 !!== 
\mathsf{y}
\mathrel {\ddot {=}}\dot{\neg}\,(
\mathsf{x}
 == 
\mathsf{y}
)n
]
\rceil 
)
]$




(*** REGELLEMMAER ***)




(*** UDSAGNSLOGIK ***)


$
[ToNegatedImply
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
]$

$
[ToNegatedImply
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\vdash RemoveDoubleNeg
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
; MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\rhd \underline{
\mathsf {a}}
\gg \underline{
\mathsf {b}}
; FromContradiction
\rhd \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
\gg \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
; \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\vdash MP2
\rhd \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
\rhd \underline{
\mathsf {a}}
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
; AutoImply
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
; Neg
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$


$
[TND
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
]$

$
[TND
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon AutoImply
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
; Repetition
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$
[FromNegations
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \underline{
\mathsf {b}}
]$

$
[FromNegations
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash TND
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
; FromDisjuncts
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\gg \underline{
\mathsf {b}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$
[
\mbox{prop lemma imply negation}
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
]$

$
[
\mbox{prop lemma imply negation}
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash AutoImply
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
; TND
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
; FromDisjuncts
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\rhd \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$
[From3Disjuncts
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {d}}
]$

$
[From3Disjuncts
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash Repetition
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
; MP
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
; FromDisjuncts
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\rhd \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\gg \underline{
\mathsf {d}}
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash \underline{
\mathsf {d}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {d}}
; AutoImply
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
; \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\vdash MP3
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {d}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\rhd \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {d}}
; MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
; FromNegations
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {d}}
\gg \underline{
\mathsf {d}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$
[NegateDisjunct1
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash \underline{
\mathsf {b}}
]$

$
[NegateDisjunct1
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash Repetition
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
; MP
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\gg \underline{
\mathsf {b}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$


$
[NegateDisjunct2
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\vdash \underline{
\mathsf {a}}
]$

$
[NegateDisjunct2
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\vdash Repetition
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
; NegativeMT
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\gg \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

(***)

$
[ExpandDisjuncts
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
]$

$
[ExpandDisjuncts
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\vdash NegateDisjunct2
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\gg \underline{
\mathsf {a}}
; NegateDisjunct2
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\rhd \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\gg \underline{
\mathsf {c}}
; JoinConjuncts
\rhd \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {c}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
; \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash MP2
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\gg \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
; Repetition
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$


$
[From2*2Disjuncts
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon 
\forall \underline{
\mathsf {e}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {e}}
]$

$
[From2*2Disjuncts
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon 
\forall \underline{
\mathsf {e}}
\colon \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {a}}
\vdash MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \underline{
\mathsf {a}}
\gg \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
; MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \underline{
\mathsf {a}}
\gg \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
; FromDisjuncts
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\rhd \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\gg \underline{
\mathsf {e}}
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon 
\forall \underline{
\mathsf {e}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash NegateDisjunct1
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\gg \underline{
\mathsf {b}}
; MP
\rhd \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \underline{
\mathsf {b}}
\gg \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
; MP
\rhd \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \underline{
\mathsf {b}}
\gg \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
; FromDisjuncts
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\rhd \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\gg \underline{
\mathsf {e}}
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon 
\forall \underline{
\mathsf {e}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon 
\forall \underline{
\mathsf {e}}
\colon \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {e}}
\gg \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {e}}
; Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon 
\forall \underline{
\mathsf {e}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash \underline{
\mathsf {e}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {e}}
; \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\vdash \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\vdash MP3
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {e}}
; MP4
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {e}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {d}}
\rhd \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {e}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {e}}
; FromNegations
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {e}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {e}}
\gg \underline{
\mathsf {e}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$














(*** SAME-F ***)
XX-am

(*** R-AFDELINGEN ***)
XX-am


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

$
[FromNegatedImply
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
]$

$
[FromNegatedImply
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\vdash \underline{
\mathsf {a}}
\vdash MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\rhd \underline{
\mathsf {a}}
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
; RemoveDoubleNeg
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\gg \underline{
\mathsf {b}}
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\gg \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
; \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
\vdash MT
\rhd \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
; Repetition
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

(***)


$
[FromNegated(2*Imply)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
)n
]$

$
[FromNegated(2*Imply)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
\vdash FromNegatedImply
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
)n
)n
; FirstConjunct
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
)n
)n
\gg \underline{
\mathsf {a}}
; SecondConjunct
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
; FromNegatedImply
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
)n
; FirstConjunct
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
)n
\gg \underline{
\mathsf {b}}
; SecondConjunct
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {c}}
)n
; JoinConjuncts
\rhd \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {b}}
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
; JoinConjuncts
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$





$
[FromNegatedOr
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
]$

$
[FromNegatedOr
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
\vdash Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
; FromNegatedImply
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$


$
[InductionAxiom
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \langle \underline{
\mathsf {b}}
{\equiv} \underline{
\mathsf {a}}
| \underline{(v1)}
{:==} 0
\rangle_{Me}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \langle \underline{
\mathsf {c}}
{\equiv} \underline{
\mathsf {a}}
| \underline{(v1)}
{:==} (\underline{(v1)}
+1
)
\rangle_{Me}
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \underline{
\mathsf {b}}
\Rightarrow 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
]
[InductionAxiom
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


$
[LessMinus1(N)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon Nat(\underline{
\mathsf {m}}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} Nat(\underline{
\mathsf {n}}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \dot{\neg}\,(\underline{
\mathsf {m}}
 <= (\underline{
\mathsf {n}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {m}}
 = (\underline{
\mathsf {n}}
+1
)
)n
)n
)n
\vdash \underline{
\mathsf {m}}
 <= \underline{
\mathsf {n}}
]
[LessMinus1(N)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[Nonnegative(N)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon Nat(\underline{
\mathsf {m}}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 0
 <= \underline{
\mathsf {m}}
]
[Nonnegative(N)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$



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

$
[Cauchy
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{(v2)}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall_{obj} \underline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon 
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{(v1)}
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{(v2)}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{(v1)}
]
+(-u\underline{(fx)}
[\underline{(v2)}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{(v1)}
]
+(-u\underline{(fx)}
[\underline{(v2)}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
]
[Cauchy
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

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





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

$
[JoinConjuncts(2conditions)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
)n
]$

$
[JoinConjuncts(2conditions)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\vdash MP2
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {b}}
\gg \underline{
\mathsf {c}}
; MP2
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\rhd \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {b}}
\gg \underline{
\mathsf {d}}
; JoinConjuncts
\rhd \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {d}}
\gg \dot{\neg}\,(\underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
)n
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
)n
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
)n
; \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\vdash MP2
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
)n
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {d}}
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {d}}
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$


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

$
[FromNegatedAnd
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\vdash \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
]$

$
[FromNegatedAnd
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\vdash \underline{
\mathsf {a}}
\vdash Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
; RemoveDoubleNeg
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\gg \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
; MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\rhd \underline{
\mathsf {a}}
\gg \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

--------


----------

$
[ToNegatedOr
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
]$

$
[ToNegatedOr
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash FirstConjunct
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
; SecondConjunct
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {b}}
)n
; NegateDisjunct1
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\gg \underline{
\mathsf {b}}
; FromContradiction
\rhd \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
; \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\vdash MP
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
; 
\mbox{prop lemma imply negation}
\rhd \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
\Rightarrow \underline{
\mathsf {b}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$




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

$
[NextXS(UpperBound)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon UB(01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
, SetOfReals
)
\vdash xs
[(\underline{
\mathsf {m}}
+1
)
]
 == xs
[\underline{
\mathsf {m}}
]
]
[NextXS(UpperBound)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[NextXS(NoUpperBound)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon \dot{\neg}\,(UB(01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
, SetOfReals
)
)n
\vdash xs
[(\underline{
\mathsf {m}}
+1
)
]
 == 01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
]
[NextXS(NoUpperBound)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[NextUS(UpperBound)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon UB(01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
, SetOfReals
)
\vdash us
[(\underline{
\mathsf {m}}
+1
)
]
 == 01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
]
[NextUS(UpperBound)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[NextUS(NoUpperBound)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon \dot{\neg}\,(UB(01//02
**xs
[\underline{
\mathsf {m}}
]
++us
[\underline{
\mathsf {m}}
]
, SetOfReals
)
)n
\vdash us
[(\underline{
\mathsf {m}}
+1
)
]
 == us
[\underline{
\mathsf {m}}
]
]
[NextUS(NoUpperBound)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$  

$
[US0
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash us
[0
]
 == xs
[0
]
++R(
\mbox{1f}
)
]
[US0
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[ExpZero
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \underline{
\mathsf {m}}
 = 0
\vdash  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 = 1
]
[ExpZero
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[ExpPositive
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf {m}}
)n
)n
)n
\vdash  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+(-u1
)
)
)
]
[ExpPositive
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

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

$
[BSzero
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \underline{
\mathsf {n}}
 = 0
\vdash BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
]
[BSzero
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[BSpositive
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf {n}}
)n
)n
)n
\vdash BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+BS(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+(-u1
)
)
)
)
]
[BSpositive
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

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

$
[UStelescope(Zero)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \underline{
\mathsf {n}}
 = 0
\vdash UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
]
[UStelescope(Zero)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[UStelescope(Positive)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf {n}}
)n
)n
)n
\vdash UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 = (|(us
[(\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+(-u1
)
)
)
)
]
[UStelescope(Positive)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


$
[(
\mathsf{x}
)
\stackrel{
\mathrm {macro}
}{\rightarrow}
\lambda 
\mathsf {t}
.
\lambda 
\mathsf {s}
.
\lambda 
\mathsf {c}
.
\tilde {{\cal M}}_4(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
,
\lceil 
[(
\mathsf{x}
)
\mathrel {\ddot {=}}
(
\mathsf{x}
)
]
\rceil 
)
]$

----------


$
[EqAddition(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon R(\underline{(fx)}
)
 = R(\underline{(fy)}
)
\vdash R(\underline{(fx)}
)
++R(\underline{(fz)}
)
 = R(\underline{(fy)}
)
++R(\underline{(fz)}
)
]
[EqAddition(R)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[PlusCommutativity(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon R(\underline{(fx)}
)
++R(\underline{(fy)}
)
 == R(\underline{(fy)}
)
++R(\underline{(fx)}
)
]
[PlusCommutativity(R)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


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

$
[PlusAssociativity(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon R(\underline{(fx)}
)
++R(\underline{(fy)}
)
++R(\underline{(fz)}
)
 = R(\underline{(fx)}
)
++R(\underline{(fy)}
)
++R(\underline{(fz)}
)
]
[PlusAssociativity(R)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[PlusAssociativity(R)XX
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon R(\underline{(fx)}
+_{f}\underline{(fy)}
+_{f}\underline{(fz)}
)
 == R(\underline{(fx)}
+_{f}\underline{(fy)}
+_{f}\underline{(fz)}
)
]
[PlusAssociativity(R)XX
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[Plus0(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon R(\underline{(fx)}
)
++R(0f
)
 == R(\underline{(fx)}
)
]
[Plus0(R)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[Negative(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(fx)}
\colon R(\underline{(fx)}
)
++(--R(\underline{(fx)}
)
)
 == R(0f
)
]
[Negative(R)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[TimesAssociativity(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon R(\underline{(fx)}
)
**R(\underline{(fy)}
)
**R(\underline{(fz)}
)
 == R(\underline{(fx)}
)
**R(\underline{(fy)}
)
**R(\underline{(fz)}
)
]
[TimesAssociativity(R)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$

$
[Times1(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon R(\underline{(fx)}
)
**R(
\mbox{1f}
)
 == R(\underline{(fx)}
)
]
[Times1(R)
\stackrel{
proof
}{\rightarrow}
Rule\ tactic
]$


$
\mathsf {a}$

venter-------





\begin {flushleft}$
[kvanti
\stackrel{
\mathrm{prio}
}{\rightarrow}
\newline \mathbf {Preassociative} \newline 
[kvanti
] , \linebreak [0] 
[
\mbox{base}
] , \linebreak [0] 
[
\mbox{bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\mbox{big bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\ \$
\ast \$\linebreak[0]\ 
] , \linebreak [0] 
[
\mathbf{flush\ left\ }[ 
\ast 
]
] , \linebreak [0] 
[
\mathsf{x}
] , \linebreak [0] 
[
\mathsf{y}
] , \linebreak [0] 
[
\mathsf{z}
] , \linebreak [0] 
[
[
\ast 
\bowtie
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel{
\ast 
}{\rightarrow}
\ast 
]
] , \linebreak [0] 
[
\mathrm{pyk}
] , \linebreak [0] 
[
\mathrm{tex}
] , \linebreak [0] 
[
\mathrm{name}
] , \linebreak [0] 
[
\mathrm{prio}
] , \linebreak [0] 
[
\ast 
] , \linebreak [0] 
[
\mathsf {T}
] , \linebreak [0] 
[
\mathrm {if}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {
\ast 
}{\Rightarrow }
\ast 
]
] , \linebreak [0] 
[
\mathrm {val}
] , \linebreak [0] 
[
\mathrm {claim}
] , \linebreak [0] 
[
\bot 
] , \linebreak [0] 
[
f(
\ast 
)
] , \linebreak [0] 
[
(
\ast 
){}^{I} 
] , \linebreak [0] 
[
\mathsf {F}
] , \linebreak [0] 
[
\underline {0}
] , \linebreak [0] 
[
\underline {1}
] , \linebreak [0] 
[
\underline {2}
] , \linebreak [0] 
[
\underline {3}
] , \linebreak [0] 
[
\underline {4}
] , \linebreak [0] 
[
\underline {5}
] , \linebreak [0] 
[
\underline {6}
] , \linebreak [0] 
[
\underline {7}
] , \linebreak [0] 
[
\underline {8}
] , \linebreak [0] 
[
\underline {9}
] , \linebreak [0] 
[
0
] , \linebreak [0] 
[
1
] , \linebreak [0] 
[
2
] , \linebreak [0] 
[
3
] , \linebreak [0] 
[
4
] , \linebreak [0] 
[
5
] , \linebreak [0] 
[
6
] , \linebreak [0] 
[
7
] , \linebreak [0] 
[
8
] , \linebreak [0] 
[
9
] , \linebreak [0] 
[
\mathsf {a}
] , \linebreak [0] 
[
\mathsf {b}
] , \linebreak [0] 
[
\mathsf {c}
] , \linebreak [0] 
[
\mathsf {d}
] , \linebreak [0] 
[
\mathsf {e}
] , \linebreak [0] 
[
\mathsf {f}
] , \linebreak [0] 
[
\mathsf {g}
] , \linebreak [0] 
[
\mathsf {h}
] , \linebreak [0] 
[
\mathsf {i}
] , \linebreak [0] 
[
\mathsf {j}
] , \linebreak [0] 
[
\mathsf {k}
] , \linebreak [0] 
[
\mathsf {l}
] , \linebreak [0] 
[
\mathsf {m}
] , \linebreak [0] 
[
\mathsf {n}
] , \linebreak [0] 
[
\mathsf {o}
] , \linebreak [0] 
[
\mathsf {p}
] , \linebreak [0] 
[
\mathsf {q}
] , \linebreak [0] 
[
\mathsf {r}
] , \linebreak [0] 
[
\mathsf {s}
] , \linebreak [0] 
[
\mathsf {t}
] , \linebreak [0] 
[
\mathsf {u}
] , \linebreak [0] 
[
\mathsf {v}
] , \linebreak [0] 
[
\mathsf {w}
] , \linebreak [0] 
[
(
\ast 
)^M
] , \linebreak [0] 
[
\mathrm {If}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[\mathrm {array}\{
\ast 
\}
\ast 
\mathrm {end\ array}
] , \linebreak [0] 
[
l
] , \linebreak [0] 
[
c
] , \linebreak [0] 
[
r
] , \linebreak [0] 
[
\mathrm {empty}
] , \linebreak [0] 
[
\langle 
\ast 
\,{|}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
{\cal M}(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal U}}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}^M(
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}_1(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
identifier( 
\ast 
)
] , \linebreak [0] 
[
identifier_{1}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]remove( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]put( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]add( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rack
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {vector}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {bibliography}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {dictionary}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {body}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {codex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {expansion}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {code}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {cache}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {diagnose}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pyk}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {tex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {texname}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {value}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {message}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {macro}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {definition}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {unpack}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {claim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {priority}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {lambda}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {apply}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {true}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {if}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {quote}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {proclaim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {define}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {introduce}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {hide}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pre}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {post}\mbox {\tt \char34}
] , \linebreak [0] 
[
{\cal E}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {lookup}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {abstract}(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\lceil 
\ast 
\rceil 
] , \linebreak [0] 
[
{\cal M}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathrm {macro}
] , \linebreak [0] 
[
s_0
] , \linebreak [0] 
[
\mathbf {zip}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {assoc}_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[(
\ast 
)^{\mathbf {p}}
] , \linebreak [0] 
[
\mbox{self}
] , \linebreak [0] 
[
[
\ast 
\mathrel {\ddot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\dot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\acute {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {pyk}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {tex}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {name}}{=} 
\ast 
]
] , \linebreak [0] 
[
\mathbf{Priority\ table} [ 
\ast 
]
] , \linebreak [0] 
[
\tilde {{\cal M}}_1
] , \linebreak [0] 
[
\tilde {{\cal M}}_2(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_3(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
(
\ast 
)
] , \linebreak [0] 
[
\left( 
\ast 
\right) 
] , \linebreak [0] 
[
display(
\ast 
)
] , \linebreak [0] 
[
statement(
\ast 
)
] , \linebreak [0] 
[
[
\ast 
]^{\cdot}
] , \linebreak [0] 
[
[
\ast 
]^-
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\langle 
\ast 
\rangle 
] , \linebreak [0] 
[
\mathbf {tuple}_1( 
\ast 
) 
] , \linebreak [0] 
[
\mathbf {tuple}_2( 
\ast 
) 
] , \linebreak [0] 
[
let_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
let_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {claim}{=}
\ast 
]
] , \linebreak [0] 
[
\mbox{checker}
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[
\mathbf{check}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_3( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\cdot} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{-} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\circ} 
] , \linebreak [0] 
[
msg
] , \linebreak [0] 
[
[
\ast 
\stackrel {msg}{=} 
\ast 
]
] , \linebreak [0] 
[
{<}stmt{>}
] , \linebreak [0] 
[
stmt
] , \linebreak [0] 
[
[
\ast 
\stackrel {stmt}{=}
\ast 
]
] , \linebreak [0] 
[
HeadNil'
] , \linebreak [0] 
[
HeadPair'
] , \linebreak [0] 
[
Transitivity'
] , \linebreak [0] 
[
{\makebox [0mm][l]{$\bot $}\,{\bot }}
] , \linebreak [0] 
[
Contra'
] , \linebreak [0] 
[
T'_{E}
] , \linebreak [0] 
[
L_{1}
] , \linebreak [0] 
[\underline{
\ast }
] , \linebreak [0] 
[{\cal A}
] , \linebreak [0] 
[{\cal B}
] , \linebreak [0] 
[{\cal C}
] , \linebreak [0] 
[{\cal D}
] , \linebreak [0] 
[{\cal E}
] , \linebreak [0] 
[{\cal F}
] , \linebreak [0] 
[{\cal G}
] , \linebreak [0] 
[{\cal H}
] , \linebreak [0] 
[{\cal I}
] , \linebreak [0] 
[{\cal J}
] , \linebreak [0] 
[{\cal K}
] , \linebreak [0] 
[{\cal L}
] , \linebreak [0] 
[{\cal M}
] , \linebreak [0] 
[{\cal N}
] , \linebreak [0] 
[{\cal O}
] , \linebreak [0] 
[{\cal P}
] , \linebreak [0] 
[{\cal Q}
] , \linebreak [0] 
[{\cal R}
] , \linebreak [0] 
[{\cal S}
] , \linebreak [0] 
[{\cal T}
] , \linebreak [0] 
[{\cal U}
] , \linebreak [0] 
[{\cal V}
] , \linebreak [0] 
[{\cal W}
] , \linebreak [0] 
[{\cal X}
] , \linebreak [0] 
[{\cal Y}
] , \linebreak [0] 
[{\cal Z}
] , \linebreak [0] 
[
\langle 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\langle ^ { \ast } 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\emptyset 
] , \linebreak [0] 
[
Remainder
] , \linebreak [0] 
[
( 
\ast 
)^{\bf v}
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error_{2}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
proof( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
proof_{2}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{I}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\rhd}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\rhd}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{E}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{E}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{+}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{+}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{-}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{-}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\ast}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\char64}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\char64}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\vdash}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\vdash}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{i.e.}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\forall}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\forall}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{;}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{;}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{;}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal T}( 
\ast 
)
] , \linebreak [0] 
[
claims( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
claims_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{<}proof{>}
] , \linebreak [0] 
[
proof
] , \linebreak [0] 
[
[ \mathbf{Lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ \mathbf{Proof\ of\ }
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antilemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ rule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antirule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
\mbox{verifier}
] , \linebreak [0] 
[
{\cal V}_1( 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_3( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_4( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_5( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_6( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_7( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Cut( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Head_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
Tail_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
rule_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rule( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Rule\ tactic
] , \linebreak [0] 
[
Plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[ \mathbf{Theory\ } 
\ast 
]
] , \linebreak [0] 
[
theory_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_3( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_4( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
HeadNil''
] , \linebreak [0] 
[
HeadPair''
] , \linebreak [0] 
[
Transitivity''
] , \linebreak [0] 
[
Contra''
] , \linebreak [0] 
[
HeadNil
] , \linebreak [0] 
[
HeadPair
] , \linebreak [0] 
[
Transitivity
] , \linebreak [0] 
[
Contra
] , \linebreak [0] 
[
T_E
] , \linebreak [0] 
[
ragged\ right
] , \linebreak [0] 
[
ragged\ right\ expansion\ 
] , \linebreak [0] 
[
parm(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
parm^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify^*(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify_2(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[L_a
] , \linebreak [0] 
[L_b
] , \linebreak [0] 
[L_c
] , \linebreak [0] 
[L_d
] , \linebreak [0] 
[L_e
] , \linebreak [0] 
[L_f
] , \linebreak [0] 
[L_g
] , \linebreak [0] 
[L_h
] , \linebreak [0] 
[L_i
] , \linebreak [0] 
[L_j
] , \linebreak [0] 
[L_k
] , \linebreak [0] 
[L_l
] , \linebreak [0] 
[L_m
] , \linebreak [0] 
[L_n
] , \linebreak [0] 
[L_o
] , \linebreak [0] 
[L_p
] , \linebreak [0] 
[L_q
] , \linebreak [0] 
[L_r
] , \linebreak [0] 
[L_s
] , \linebreak [0] 
[L_t
] , \linebreak [0] 
[L_u
] , \linebreak [0] 
[L_v
] , \linebreak [0] 
[L_w
] , \linebreak [0] 
[L_x
] , \linebreak [0] 
[L_y
] , \linebreak [0] 
[L_z
] , \linebreak [0] 
[L_A
] , \linebreak [0] 
[L_B
] , \linebreak [0] 
[L_C
] , \linebreak [0] 
[L_D
] , \linebreak [0] 
[L_E
] , \linebreak [0] 
[L_F
] , \linebreak [0] 
[L_G
] , \linebreak [0] 
[L_H
] , \linebreak [0] 
[L_I
] , \linebreak [0] 
[L_J
] , \linebreak [0] 
[L_K
] , \linebreak [0] 
[L_L
] , \linebreak [0] 
[L_M
] , \linebreak [0] 
[L_N
] , \linebreak [0] 
[L_O
] , \linebreak [0] 
[L_P
] , \linebreak [0] 
[L_Q
] , \linebreak [0] 
[L_R
] , \linebreak [0] 
[L_S
] , \linebreak [0] 
[L_T
] , \linebreak [0] 
[L_U
] , \linebreak [0] 
[L_V
] , \linebreak [0] 
[L_W
] , \linebreak [0] 
[L_X
] , \linebreak [0] 
[L_Y
] , \linebreak [0] 
[L_Z
] , \linebreak [0] 
[L_?
] , \linebreak [0] 
[
Reflexivity
] , \linebreak [0] 
[
Reflexivity_1
] , \linebreak [0] 
[
Commutativity
] , \linebreak [0] 
[
Commutativity_1
] , \linebreak [0] 
[
{<}tactic{>}
] , \linebreak [0] 
[
tactic
] , \linebreak [0] 
[
[
\ast 
\stackrel {tactic}{=}
\ast 
]
] , \linebreak [0] 
[
{\cal P}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal P}^*( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
p_0
] , \linebreak [0] 
[
conclude_1 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_2 ( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_3 ( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_4 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mbox{check}
] , \linebreak [0] 
[
[
\ast 
\stackrel {\circ}{=}
\ast 
]
] , \linebreak [0] 
[
RootVisible(
\ast 
)
] , \linebreak [0] 
[
A
] , \linebreak [0] 
[
R
] , \linebreak [0] 
[
C
] , \linebreak [0] 
[
T
] , \linebreak [0] 
[
L
] , \linebreak [0] 
[
\{
\ast 
\}
] , \linebreak [0] 
[\overline{
\ast }
] , \linebreak [0] 
[
\mathit{a}
] , \linebreak [0] 
[
\mathit{b}
] , \linebreak [0] 
[
\mathit{c}
] , \linebreak [0] 
[
\mathit{d}
] , \linebreak [0] 
[
\mathit{e}
] , \linebreak [0] 
[
\mathit{f}
] , \linebreak [0] 
[
\mathit{g}
] , \linebreak [0] 
[
\mathit{h}
] , \linebreak [0] 
[
\mathit{i}
] , \linebreak [0] 
[
\mathit{j}
] , \linebreak [0] 
[
\mathit{k}
] , \linebreak [0] 
[
\mathit{l}
] , \linebreak [0] 
[
\mathit{m}
] , \linebreak [0] 
[
\mathit{n}
] , \linebreak [0] 
[
\mathit{o}
] , \linebreak [0] 
[
\mathit{p}
] , \linebreak [0] 
[
\mathit{q}
] , \linebreak [0] 
[
\mathit{r}
] , \linebreak [0] 
[
\mathit{s}
] , \linebreak [0] 
[
\mathit{t}
] , \linebreak [0] 
[
\mathit{u}
] , \linebreak [0] 
[
\mathit{v}
] , \linebreak [0] 
[
\mathit{w}
] , \linebreak [0] 
[
\mathit{x}
] , \linebreak [0] 
[
\mathit{y}
] , \linebreak [0] 
[
\mathit{z}
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
Ded(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_0(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_7(
\ast 
)
] , \linebreak [0] 
[
Ded_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
S
] , \linebreak [0] 
[
Neg
] , \linebreak [0] 
[
MP
] , \linebreak [0] 
[
Gen
] , \linebreak [0] 
[
Ded
] , \linebreak [0] 
[
S1
] , \linebreak [0] 
[
S2
] , \linebreak [0] 
[
S3
] , \linebreak [0] 
[
S4
] , \linebreak [0] 
[
S5
] , \linebreak [0] 
[
S6
] , \linebreak [0] 
[
S7
] , \linebreak [0] 
[
S8
] , \linebreak [0] 
[
S9
] , \linebreak [0] 
[
Repetition
] , \linebreak [0] 
[
A1'
] , \linebreak [0] 
[
A2'
] , \linebreak [0] 
[
A4'
] , \linebreak [0] 
[
A5'
] , \linebreak [0] 
[
Prop\ 3.2a
] , \linebreak [0] 
[
Prop\ 3.2b
] , \linebreak [0] 
[
Prop\ 3.2c
] , \linebreak [0] 
[
Prop\ 3.2d
] , \linebreak [0] 
[
Prop\ 3.2e_1
] , \linebreak [0] 
[
Prop\ 3.2e_2
] , \linebreak [0] 
[
Prop\ 3.2e
] , \linebreak [0] 
[
Prop\ 3.2f_1
] , \linebreak [0] 
[
Prop\ 3.2f_2
] , \linebreak [0] 
[
Prop\ 3.2f
] , \linebreak [0] 
[
Prop\ 3.2g_1
] , \linebreak [0] 
[
Prop\ 3.2g_2
] , \linebreak [0] 
[
Prop\ 3.2g
] , \linebreak [0] 
[
Prop\ 3.2h_1
] , \linebreak [0] 
[
Prop\ 3.2h_2
] , \linebreak [0] 
[
Prop\ 3.2h
] , \linebreak [0] 
[
Block_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Block_2(
\ast 
)
] , \linebreak [0] 
[UniqueMember
] , \linebreak [0] 
[UniqueMember(Type)
] , \linebreak [0] 
[SameSeries
] , \linebreak [0] 
[A4
] , \linebreak [0] 
[SameMember
] , \linebreak [0] 
[Qclosed(Addition)
] , \linebreak [0] 
[Qclosed(Multiplication)
] , \linebreak [0] 
[FromCartProd(1)
] , \linebreak [0] 
[
\mbox{1rule fromCartProd(2)}
] , \linebreak [0] 
[constantRationalSeries(
\ast 
)
] , \linebreak [0] 
[cartProd(
\ast 
)
] , \linebreak [0] 
[Power(
\ast 
)
] , \linebreak [0] 
[binaryUnion(
\ast 
,
\ast 
)
] , \linebreak [0] 
[SetOfRationalSeries
] , \linebreak [0] 
[IsSubset(
\ast 
,
\ast 
)
] , \linebreak [0] 
[(p
\ast 
,
\ast 
)
] , \linebreak [0] 
[(s
\ast 
)
] , \linebreak [0] 
[(\cdots{})
] , \linebreak [0] 
[\texttt{Objekt-var}
] , \linebreak [0] 
[\texttt{Ex-var}
] , \linebreak [0] 
[\texttt{Ph-var}
] , \linebreak [0] 
[\texttt{V\ae{}rdi}
] , \linebreak [0] 
[\texttt{Variabel}
] , \linebreak [0] 
[Op(
\ast 
)
] , \linebreak [0] 
[Op(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
 \mathrel {\ddot {==}} 
\ast 
] , \linebreak [0] 
[ContainsEmpty(
\ast 
)
] , \linebreak [0] 
[Nat(
\ast 
)
] , \linebreak [0] 
[
Dedu(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_0(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Dedu_{s}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_7(
\ast 
)
] , \linebreak [0] 
[
Dedu_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Ex_{1}
] , \linebreak [0] 
[Ex_{2}
] , \linebreak [0] 
[Ex3
] , \linebreak [0] 
[Ex_{10}
] , \linebreak [0] 
[Ex_{20}
] , \linebreak [0] 
[
\ast 
 _{Ex}
] , \linebreak [0] 
[
\ast 
 ^{Ex}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[ph_{1}
] , \linebreak [0] 
[ph_{2}
] , \linebreak [0] 
[ph_{3}
] , \linebreak [0] 
[
\ast 
 _{Ph} 
] , \linebreak [0] 
[
\ast 
 ^{Ph}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me} 
] , \linebreak [0] 
[\mathsf {bs}
] , \linebreak [0] 
[ \mathsf {OBS}
] , \linebreak [0] 
[{\cal BS}
] , \linebreak [0] 
[\mathrm{\O}
] , \linebreak [0] 
[SystemQ
] , \linebreak [0] 
[MP
] , \linebreak [0] 
[Gen
] , \linebreak [0] 
[Repetition
] , \linebreak [0] 
[Neg
] , \linebreak [0] 
[Ded
] , \linebreak [0] 
[ExistIntro
] , \linebreak [0] 
[Extensionality
] , \linebreak [0] 
[\O{}def
] , \linebreak [0] 
[PairDef
] , \linebreak [0] 
[UnionDef
] , \linebreak [0] 
[PowerDef
] , \linebreak [0] 
[SeparationDef
] , \linebreak [0] 
[AddDoubleNeg
] , \linebreak [0] 
[RemoveDoubleNeg
] , \linebreak [0] 
[AndCommutativity
] , \linebreak [0] 
[AutoImply
] , \linebreak [0] 
[Contrapositive
] , \linebreak [0] 
[FirstConjunct
] , \linebreak [0] 
[SecondConjunct
] , \linebreak [0] 
[FromContradiction
] , \linebreak [0] 
[FromDisjuncts
] , \linebreak [0] 
[IffCommutativity
] , \linebreak [0] 
[IffFirst
] , \linebreak [0] 
[IffSecond
] , \linebreak [0] 
[ImplyTransitivity
] , \linebreak [0] 
[JoinConjuncts
] , \linebreak [0] 
[MP2
] , \linebreak [0] 
[MP3
] , \linebreak [0] 
[MP4
] , \linebreak [0] 
[MP5
] , \linebreak [0] 
[MT
] , \linebreak [0] 
[NegativeMT
] , \linebreak [0] 
[Technicality
] , \linebreak [0] 
[Weakening
] , \linebreak [0] 
[WeakenOr1
] , \linebreak [0] 
[WeakenOr2
] , \linebreak [0] 
[Formula2Pair
] , \linebreak [0] 
[Pair2Formula
] , \linebreak [0] 
[Formula2Union
] , \linebreak [0] 
[Union2Formula
] , \linebreak [0] 
[Formula2Sep
] , \linebreak [0] 
[Sep2Formula
] , \linebreak [0] 
[Formula2Power
] , \linebreak [0] 
[SubsetInPower
] , \linebreak [0] 
[HelperPowerIsSub
] , \linebreak [0] 
[PowerIsSub
] , \linebreak [0] 
[(Switch)HelperPowerIsSub
] , \linebreak [0] 
[(Switch)PowerIsSub
] , \linebreak [0] 
[ToSetEquality
] , \linebreak [0] 
[HelperToSetEquality(t)
] , \linebreak [0] 
[ToSetEquality(t)
] , \linebreak [0] 
[HelperFromSetEquality
] , \linebreak [0] 
[FromSetEquality
] , \linebreak [0] 
[HelperReflexivity
] , \linebreak [0] 
[Reflexivity
] , \linebreak [0] 
[HelperSymmetry
] , \linebreak [0] 
[Symmetry
] , \linebreak [0] 
[HelperTransitivity
] , \linebreak [0] 
[Transitivity
] , \linebreak [0] 
[ERisReflexive
] , \linebreak [0] 
[ERisSymmetric
] , \linebreak [0] 
[ERisTransitive
] , \linebreak [0] 
[\O{}isSubset
] , \linebreak [0] 
[HelperMemberNot\O{}
] , \linebreak [0] 
[MemberNot\O{}
] , \linebreak [0] 
[HelperUnique\O{}
] , \linebreak [0] 
[Unique\O{}
] , \linebreak [0] 
[==\!{}Reflexivity
] , \linebreak [0] 
[==\!{}Symmetry
] , \linebreak [0] 
[Helper\!{}==\!{}Transitivity
] , \linebreak [0] 
[\!{}==\!{}Transitivity
] , \linebreak [0] 
[HelperTransferNotEq
] , \linebreak [0] 
[TransferNotEq
] , \linebreak [0] 
[HelperPairSubset
] , \linebreak [0] 
[Helper(2)PairSubset
] , \linebreak [0] 
[PairSubset
] , \linebreak [0] 
[SamePair
] , \linebreak [0] 
[SameSingleton
] , \linebreak [0] 
[UnionSubset
] , \linebreak [0] 
[SameUnion
] , \linebreak [0] 
[SeparationSubset
] , \linebreak [0] 
[SameSeparation
] , \linebreak [0] 
[SameBinaryUnion
] , \linebreak [0] 
[IntersectionSubset
] , \linebreak [0] 
[SameIntersection
] , \linebreak [0] 
[AutoMember
] , \linebreak [0] 
[HelperEqSysNot\O{}
] , \linebreak [0] 
[EqSysNot\O{}
] , \linebreak [0] 
[HelperEqSubset
] , \linebreak [0] 
[EqSubset
] , \linebreak [0] 
[HelperEqNecessary
] , \linebreak [0] 
[EqNecessary
] , \linebreak [0] 
[HelperNoneEqNecessary
] , \linebreak [0] 
[Helper(2)NoneEqNecessary
] , \linebreak [0] 
[NoneEqNecessary
] , \linebreak [0] 
[EqClassIsSubset
] , \linebreak [0] 
[EqClassesAreDisjoint
] , \linebreak [0] 
[AllDisjoint
] , \linebreak [0] 
[AllDisjointImply
] , \linebreak [0] 
[BSsubset
] , \linebreak [0] 
[Union(BS/R)subset
] , \linebreak [0] 
[UnionIdentity
] , \linebreak [0] 
[EqSysIsPartition
] , \linebreak [0] 
[(x1)
] , \linebreak [0] 
[(x2)
] , \linebreak [0] 
[(y1)
] , \linebreak [0] 
[(y2)
] , \linebreak [0] 
[(v1)
] , \linebreak [0] 
[(v2)
] , \linebreak [0] 
[(v3)
] , \linebreak [0] 
[(v4)
] , \linebreak [0] 
[(v2n)
] , \linebreak [0] 
[(m1)
] , \linebreak [0] 
[(m2)
] , \linebreak [0] 
[(n1)
] , \linebreak [0] 
[(n2)
] , \linebreak [0] 
[(n3)
] , \linebreak [0] 
[(\epsilon)
] , \linebreak [0] 
[(\epsilon)_{1}
] , \linebreak [0] 
[(\epsilon 2)
] , \linebreak [0] 
[(fep)
] , \linebreak [0] 
[(fx)
] , \linebreak [0] 
[(fy)
] , \linebreak [0] 
[(fz)
] , \linebreak [0] 
[(fu)
] , \linebreak [0] 
[(fv)
] , \linebreak [0] 
[(fw)
] , \linebreak [0] 
[(rx)
] , \linebreak [0] 
[(ry)
] , \linebreak [0] 
[(rz)
] , \linebreak [0] 
[(ru)
] , \linebreak [0] 
[(sx)
] , \linebreak [0] 
[(sx1)
] , \linebreak [0] 
[(sy)
] , \linebreak [0] 
[(sy1)
] , \linebreak [0] 
[(sz)
] , \linebreak [0] 
[(sz1)
] , \linebreak [0] 
[(su)
] , \linebreak [0] 
[(su1)
] , \linebreak [0] 
[(fxs)
] , \linebreak [0] 
[(fys)
] , \linebreak [0] 
[(crs1)
] , \linebreak [0] 
[(f1)
] , \linebreak [0] 
[(f2)
] , \linebreak [0] 
[(f3)
] , \linebreak [0] 
[(f4)
] , \linebreak [0] 
[(op1)
] , \linebreak [0] 
[(op2)
] , \linebreak [0] 
[(r1)
] , \linebreak [0] 
[(s1)
] , \linebreak [0] 
[(s2)
] , \linebreak [0] 
[X_{1}
] , \linebreak [0] 
[X_{2}
] , \linebreak [0] 
[Y_{1}
] , \linebreak [0] 
[Y_{2}
] , \linebreak [0] 
[V_{1}
] , \linebreak [0] 
[V_{2}
] , \linebreak [0] 
[V_{3}
] , \linebreak [0] 
[V_{4}
] , \linebreak [0] 
[V_{2n}
] , \linebreak [0] 
[M_{1}
] , \linebreak [0] 
[M_{2}
] , \linebreak [0] 
[N_{1} 
] , \linebreak [0] 
[N_{2} 
] , \linebreak [0] 
[N_{3} 
] , \linebreak [0] 
[\epsilon
] , \linebreak [0] 
[\epsilon 1
] , \linebreak [0] 
[\epsilon 2
] , \linebreak [0] 
[FX
] , \linebreak [0] 
[FY
] , \linebreak [0] 
[FZ
] , \linebreak [0] 
[FU
] , \linebreak [0] 
[FV
] , \linebreak [0] 
[FW
] , \linebreak [0] 
[FEP
] , \linebreak [0] 
[RX
] , \linebreak [0] 
[RY
] , \linebreak [0] 
[RZ
] , \linebreak [0] 
[RU
] , \linebreak [0] 
[(SX)
] , \linebreak [0] 
[(SX1)
] , \linebreak [0] 
[(SY)
] , \linebreak [0] 
[(SY1)
] , \linebreak [0] 
[(SZ)
] , \linebreak [0] 
[(SZ1)
] , \linebreak [0] 
[(SU)
] , \linebreak [0] 
[(SU1)
] , \linebreak [0] 
[FXS
] , \linebreak [0] 
[FYS
] , \linebreak [0] 
[(F1)
] , \linebreak [0] 
[(F2)
] , \linebreak [0] 
[(F3)
] , \linebreak [0] 
[(F4)
] , \linebreak [0] 
[(OP1)
] , \linebreak [0] 
[(OP2)
] , \linebreak [0] 
[(R1)
] , \linebreak [0] 
[(S1)
] , \linebreak [0] 
[(S2)
] , \linebreak [0] 
[(EPob)
] , \linebreak [0] 
[(CRS1ob)
] , \linebreak [0] 
[(F1ob)
] , \linebreak [0] 
[(F2ob)
] , \linebreak [0] 
[(F3ob)
] , \linebreak [0] 
[(F4ob)
] , \linebreak [0] 
[(N1ob)
] , \linebreak [0] 
[(N2ob)
] , \linebreak [0] 
[(OP1ob)
] , \linebreak [0] 
[(OP2ob)
] , \linebreak [0] 
[(R1ob)
] , \linebreak [0] 
[(S1ob)
] , \linebreak [0] 
[(S2ob)
] , \linebreak [0] 
[ph_{4}
] , \linebreak [0] 
[ph_{5}
] , \linebreak [0] 
[ph_{6}
] , \linebreak [0] 
[NAT
] , \linebreak [0] 
[RATIONAL_SERIES
] , \linebreak [0] 
[SERIES
] , \linebreak [0] 
[SetOfReals
] , \linebreak [0] 
[SetOfFxs
] , \linebreak [0] 
[N
] , \linebreak [0] 
[Q
] , \linebreak [0] 
[X
] , \linebreak [0] 
[xs
] , \linebreak [0] 
[xaF
] , \linebreak [0] 
[ysF
] , \linebreak [0] 
[us
] , \linebreak [0] 
[usFoelge
] , \linebreak [0] 
[0
] , \linebreak [0] 
[1
] , \linebreak [0] 
[(-1)
] , \linebreak [0] 
[2
] , \linebreak [0] 
[3
] , \linebreak [0] 
[1/2
] , \linebreak [0] 
[1/3
] , \linebreak [0] 
[2/3
] , \linebreak [0] 
[0f
] , \linebreak [0] 
[
\mbox{1f}
] , \linebreak [0] 
[00
] , \linebreak [0] 
[
\mbox{01}
] , \linebreak [0] 
[(--01)
] , \linebreak [0] 
[02
] , \linebreak [0] 
[01//02
] , \linebreak [0] 
[PlusAssociativity(R)
] , \linebreak [0] 
[PlusAssociativity(R)XX
] , \linebreak [0] 
[Plus0(R)
] , \linebreak [0] 
[Negative(R)
] , \linebreak [0] 
[Times1(R)
] , \linebreak [0] 
[lessAddition(R)
] , \linebreak [0] 
[PlusCommutativity(R)
] , \linebreak [0] 
[LeqAntisymmetry(R)
] , \linebreak [0] 
[LeqTransitivity(R)
] , \linebreak [0] 
[leqAddition(R)
] , \linebreak [0] 
[Distribution(R)
] , \linebreak [0] 
[A4(Axiom)
] , \linebreak [0] 
[InductionAxiom
] , \linebreak [0] 
[EqualityAxiom
] , \linebreak [0] 
[EqLeqAxiom
] , \linebreak [0] 
[EqAdditionAxiom
] , \linebreak [0] 
[EqMultiplicationAxiom
] , \linebreak [0] 
[QisClosed(Reciprocal)(Imply)
] , \linebreak [0] 
[QisClosed(Reciprocal)
] , \linebreak [0] 
[QisClosed(Negative)(Imply)
] , \linebreak [0] 
[QisClosed(Negative)
] , \linebreak [0] 
[leqReflexivity
] , \linebreak [0] 
[leqAntisymmetryAxiom
] , \linebreak [0] 
[leqTransitivityAxiom
] , \linebreak [0] 
[leqTotality
] , \linebreak [0] 
[leqAdditionAxiom
] , \linebreak [0] 
[leqMultiplicationAxiom
] , \linebreak [0] 
[plusAssociativity
] , \linebreak [0] 
[plusCommutativity
] , \linebreak [0] 
[Negative
] , \linebreak [0] 
[plus0
] , \linebreak [0] 
[timesAssociativity
] , \linebreak [0] 
[timesCommutativity
] , \linebreak [0] 
[ReciprocalAxiom
] , \linebreak [0] 
[times1
] , \linebreak [0] 
[Distribution
] , \linebreak [0] 
[0not1
] , \linebreak [0] 
[
\mbox{lemma eqLeq(R)}
] , \linebreak [0] 
[TimesAssociativity(R)
] , \linebreak [0] 
[TimesCommutativity(R)
] , \linebreak [0] 
[(Adgic)SameR
] , \linebreak [0] 
[Separation2formula(1)
] , \linebreak [0] 
[Separation2formula(2)
] , \linebreak [0] 
[Cauchy
] , \linebreak [0] 
[PlusF
] , \linebreak [0] 
[ReciprocalF
] , \linebreak [0] 
[From==
] , \linebreak [0] 
[To==
] , \linebreak [0] 
[FromInR
] , \linebreak [0] 
[PlusR(Sym)
] , \linebreak [0] 
[ReciprocalR(Axiom)
] , \linebreak [0] 
[LessMinus1(N)
] , \linebreak [0] 
[Nonnegative(N)
] , \linebreak [0] 
[US0
] , \linebreak [0] 
[NextXS(UpperBound)
] , \linebreak [0] 
[NextXS(NoUpperBound)
] , \linebreak [0] 
[NextUS(UpperBound)
] , \linebreak [0] 
[NextUS(NoUpperBound)
] , \linebreak [0] 
[ExpZero
] , \linebreak [0] 
[ExpPositive
] , \linebreak [0] 
[ExpZero(R)
] , \linebreak [0] 
[ExpPositive(R)
] , \linebreak [0] 
[BSzero
] , \linebreak [0] 
[BSpositive
] , \linebreak [0] 
[UStelescope(Zero)
] , \linebreak [0] 
[UStelescope(Positive)
] , \linebreak [0] 
[EqAddition(R)
] , \linebreak [0] 
[FromLimit
] , \linebreak [0] 
[ToUpperBound
] , \linebreak [0] 
[FromUpperBound
] , \linebreak [0] 
[USisUpperBound
] , \linebreak [0] 
[0not1(R)
] , \linebreak [0] 
[ExpUnbounded(R)
] , \linebreak [0] 
[FromLeq(Advanced)(N)
] , \linebreak [0] 
[FromLeastUpperBound
] , \linebreak [0] 
[ToLeastUpperBound
] , \linebreak [0] 
[XSisNotUpperBound
] , \linebreak [0] 
[ysFGreater
] , \linebreak [0] 
[ysFLess
] , \linebreak [0] 
[SmallInverse
] , \linebreak [0] 
[NatType
] , \linebreak [0] 
[RationalType
] , \linebreak [0] 
[SeriesType
] , \linebreak [0] 
[Max
] , \linebreak [0] 
[Numerical
] , \linebreak [0] 
[NumericalF
] , \linebreak [0] 
[MemberOfSeries(Imply)
] , \linebreak [0] 
[JoinConjuncts(2conditions)
] , \linebreak [0] 
[
\mbox{prop lemma imply negation}
] , \linebreak [0] 
[TND
] , \linebreak [0] 
[FromNegatedImply
] , \linebreak [0] 
[ToNegatedImply
] , \linebreak [0] 
[FromNegated(2*Imply)
] , \linebreak [0] 
[FromNegatedAnd
] , \linebreak [0] 
[FromNegatedOr
] , \linebreak [0] 
[ToNegatedOr
] , \linebreak [0] 
[FromNegations
] , \linebreak [0] 
[From3Disjuncts
] , \linebreak [0] 
[From2*2Disjuncts
] , \linebreak [0] 
[NegateDisjunct1
] , \linebreak [0] 
[NegateDisjunct2
] , \linebreak [0] 
[ExpandDisjuncts
] , \linebreak [0] 
[SENC1
] , \linebreak [0] 
[SENC2
] , \linebreak [0] 
[LessLeq(R)
] , \linebreak [0] 
[MemberOfSeries
] , \linebreak [0] 
[memberOfSeries(Type)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\_\{
\ast 
\}
] , \linebreak [0] 
[
\ast 
/indexintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/bothintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/nameintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast '
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\Rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
0
] , \linebreak [0] 
[
\ast 
1
] , \linebreak [0] 
[
0 \mathrm {b}
] , \linebreak [0] 
[
\ast 
\mbox {-color}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
\mbox {-color}^{\ast}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
{}^H
] , \linebreak [0] 
[
\ast 
{}^T
] , \linebreak [0] 
[
\ast 
{}^U
] , \linebreak [0] 
[
\ast 
{}^h
] , \linebreak [0] 
[
\ast 
{}^t
] , \linebreak [0] 
[
\ast 
{}^s
] , \linebreak [0] 
[
\ast 
{}^c
] , \linebreak [0] 
[
\ast 
{}^d
] , \linebreak [0] 
[
\ast 
{}^a
] , \linebreak [0] 
[
\ast 
{}^C
] , \linebreak [0] 
[
\ast 
{}^M
] , \linebreak [0] 
[
\ast 
{}^B
] , \linebreak [0] 
[
\ast 
{}^{r}
] , \linebreak [0] 
[
\ast 
{}^{i}
] , \linebreak [0] 
[
\ast 
{}^{d}
] , \linebreak [0] 
[
\ast 
{}^{R}
] , \linebreak [0] 
[
\ast 
{}^{0}
] , \linebreak [0] 
[
\ast 
{}^{1}
] , \linebreak [0] 
[
\ast 
{}^{2}
] , \linebreak [0] 
[
\ast 
{}^{3}
] , \linebreak [0] 
[
\ast 
{}^{4}
] , \linebreak [0] 
[
\ast 
{}^{5}
] , \linebreak [0] 
[
\ast 
{}^{6}
] , \linebreak [0] 
[
\ast 
{}^{7}
] , \linebreak [0] 
[
\ast 
{}^{8}
] , \linebreak [0] 
[
\ast 
{}^{9}
] , \linebreak [0] 
[
\ast 
{} ^ { E }
] , \linebreak [0] 
[
\ast 
{} ^ {\cal V}
] , \linebreak [0] 
[
\ast 
{} ^ {\cal C}
] , \linebreak [0] 
[
\ast 
{} ^ {{\cal C} ^ { \ast }}
] , \linebreak [0] 
[
\ast 
{}^{hide}
]
; 
\newline \mathbf {Preassociative} \newline 
[
\mbox{``}
\ast 
\mbox{''}
] , \linebreak [0] 
[
] , \linebreak [0] 
[
(
\ast 
)^{\bf t}
] , \linebreak [0] 
[
\mbox{string}(
\ast 
)+
\ast 
] , \linebreak [0] 
[\mbox{string}(
\ast 
)\mathrel{++}
\ast 
] , \linebreak [0] 
[
\newline 
\ast 
] , \linebreak [0] 
[
\linebreak [0]\ \hskip0em plus2.0em{}
\ast 
] , \linebreak [0] 
[!
\ast 
] , \linebreak [0] 
[
\mbox {\tt \char34}
\ast 
] , \linebreak [0] 
[
\#
\ast 
] , \linebreak [0] 
[
\$
\ast 
] , \linebreak [0] 
[
\%
\ast 
] , \linebreak [0] 
[
\&
\ast 
] , \linebreak [0] 
[
\mbox {'}
\ast 
] , \linebreak [0] 
[(
\ast 
] , \linebreak [0] 
[)
\ast 
] , \linebreak [0] 
[
{*}
\ast 
] , \linebreak [0] 
[
{+}
\ast 
] , \linebreak [0] 
[,
\ast 
] , \linebreak [0] 
[
\mbox{-}
\ast 
] , \linebreak [0] 
[.
\ast 
] , \linebreak [0] 
[/
\ast 
] , \linebreak [0] 
[0
\ast 
] , \linebreak [0] 
[1
\ast 
] , \linebreak [0] 
[2
\ast 
] , \linebreak [0] 
[3
\ast 
] , \linebreak [0] 
[4
\ast 
] , \linebreak [0] 
[5
\ast 
] , \linebreak [0] 
[6
\ast 
] , \linebreak [0] 
[7
\ast 
] , \linebreak [0] 
[8
\ast 
] , \linebreak [0] 
[9
\ast 
] , \linebreak [0] 
[
{:}
\ast 
] , \linebreak [0] 
[;
\ast 
] , \linebreak [0] 
[
{<}
\ast 
] , \linebreak [0] 
[
{=}
\ast 
] , \linebreak [0] 
[
{>}
\ast 
] , \linebreak [0] 
[?
\ast 
] , \linebreak [0] 
[@
\ast 
] , \linebreak [0] 
[A
\ast 
] , \linebreak [0] 
[B
\ast 
] , \linebreak [0] 
[C
\ast 
] , \linebreak [0] 
[D
\ast 
] , \linebreak [0] 
[E
\ast 
] , \linebreak [0] 
[F
\ast 
] , \linebreak [0] 
[G
\ast 
] , \linebreak [0] 
[H
\ast 
] , \linebreak [0] 
[I
\ast 
] , \linebreak [0] 
[J
\ast 
] , \linebreak [0] 
[K
\ast 
] , \linebreak [0] 
[L
\ast 
] , \linebreak [0] 
[M
\ast 
] , \linebreak [0] 
[N
\ast 
] , \linebreak [0] 
[O
\ast 
] , \linebreak [0] 
[P
\ast 
] , \linebreak [0] 
[Q
\ast 
] , \linebreak [0] 
[R
\ast 
] , \linebreak [0] 
[S
\ast 
] , \linebreak [0] 
[T
\ast 
] , \linebreak [0] 
[U
\ast 
] , \linebreak [0] 
[V
\ast 
] , \linebreak [0] 
[W
\ast 
] , \linebreak [0] 
[X
\ast 
] , \linebreak [0] 
[Y
\ast 
] , \linebreak [0] 
[Z
\ast 
] , \linebreak [0] 
[[
\ast 
] , \linebreak [0] 
[
\mbox {$\backslash $}
\ast 
] , \linebreak [0] 
[]
\ast 
] , \linebreak [0] 
[
{\char94}
\ast 
] , \linebreak [0] 
[
\_
\ast 
] , \linebreak [0] 
[
\mbox {`}
\ast 
] , \linebreak [0] 
[a
\ast 
] , \linebreak [0] 
[b
\ast 
] , \linebreak [0] 
[c
\ast 
] , \linebreak [0] 
[d
\ast 
] , \linebreak [0] 
[e
\ast 
] , \linebreak [0] 
[f
\ast 
] , \linebreak [0] 
[g
\ast 
] , \linebreak [0] 
[h
\ast 
] , \linebreak [0] 
[i
\ast 
] , \linebreak [0] 
[j
\ast 
] , \linebreak [0] 
[k
\ast 
] , \linebreak [0] 
[l
\ast 
] , \linebreak [0] 
[m
\ast 
] , \linebreak [0] 
[n
\ast 
] , \linebreak [0] 
[o
\ast 
] , \linebreak [0] 
[p
\ast 
] , \linebreak [0] 
[q
\ast 
] , \linebreak [0] 
[r
\ast 
] , \linebreak [0] 
[s
\ast 
] , \linebreak [0] 
[t
\ast 
] , \linebreak [0] 
[u
\ast 
] , \linebreak [0] 
[v
\ast 
] , \linebreak [0] 
[w
\ast 
] , \linebreak [0] 
[x
\ast 
] , \linebreak [0] 
[y
\ast 
] , \linebreak [0] 
[z
\ast 
] , \linebreak [0] 
[
\{
\ast 
] , \linebreak [0] 
[|
\ast 
] , \linebreak [0] 
[
\}
\ast 
] , \linebreak [0] 
[
\char126 
\ast 
] , \linebreak [0] 
[
\mathbf{Preassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
\mathbf{Postassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
[
\ast 
] , \linebreak [0] 
\ast 
] , \linebreak [0] 
[
\mathrm{priority} \,
\ast 
\, \mathrm{end}
] , \linebreak [0] 
[
newline\ 
\ast 
] , \linebreak [0] 
[
macro\ newline\ 
\ast 
] , \linebreak [0] 
[
MacroIndent(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbin {\mbox {'}}
\ast 
] , \linebreak [0] 
[
\ast 
\mathbin {\mbox {`}}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[ 
\ast 
 (exp) 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{}'
] , \linebreak [0] 
[R(
\ast 
)
] , \linebreak [0] 
[--R(
\ast 
)
] , \linebreak [0] 
[rec
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 / 
\ast 
] , \linebreak [0] 
[
\ast 
 \cap 
\ast 
] , \linebreak [0] 
[
\ast 
[
\ast 
]
]
; 
\newline \mathbf {Preassociative} \newline 
[\cup 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\cup} 
\ast 
] , \linebreak [0] 
[P(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[\{
\ast 
\}
] , \linebreak [0] 
[StateExpand(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[extractSeries(
\ast 
)
] , \linebreak [0] 
[SetOfSeries(
\ast 
)
] , \linebreak [0] 
[--Macro(
\ast 
)
] , \linebreak [0] 
[ExpandList(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[**Macro(
\ast 
)
] , \linebreak [0] 
[++Macro(
\ast 
)
] , \linebreak [0] 
[<<Macro(
\ast 
)
] , \linebreak [0] 
[||Macro(
\ast 
)
] , \linebreak [0] 
[01//Macro(
\ast 
)
] , \linebreak [0] 
[UB(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[LUB(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[BS(
\ast 
,
\ast 
)
] , \linebreak [0] 
[UStelescope(
\ast 
,
\ast 
)
] , \linebreak [0] 
[(
\ast 
)
] , \linebreak [0] 
[|f
\ast 
|
] , \linebreak [0] 
[|r
\ast 
|
] , \linebreak [0] 
[Limit(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Union(
\ast 
)
] , \linebreak [0] 
[IsOrderedPair(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsRelation(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[isFunction(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsSeries(
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsNatural(
\ast 
,\ast
)
] , \linebreak [0] 
[OrderedPair(
\ast 
,
\ast 
)
] , \linebreak [0] 
[TypeNat(
\ast 
)
] , \linebreak [0] 
[TypeNat0(
\ast 
)
] , \linebreak [0] 
[TypeRational(
\ast 
)
] , \linebreak [0] 
[TypeRational0(
\ast 
)
] , \linebreak [0] 
[TypeSeries(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Typeseries0(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[\{
\ast 
,
\ast 
\}
] , \linebreak [0] 
[\langle 
\ast 
,
\ast 
\rangle
] , \linebreak [0] 
[(-u
\ast 
)
] , \linebreak [0] 
[-_{f}
\ast 
] , \linebreak [0] 
[(--
\ast 
)
] , \linebreak [0] 
[1f/
\ast 
] , \linebreak [0] 
[01//temp
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
(
\ast 
,
\ast 
)
] , \linebreak [0] 
[ReflRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[SymRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[TransRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[EqRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[[
\ast 
 \mathrel{\in} 
\ast 
]_{
\ast 
}
] , \linebreak [0] 
[Partition(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cdot 
\ast 
] , \linebreak [0] 
[
\ast 
\cdot_0 
\ast 
] , \linebreak [0] 
[(
\ast 
*
\ast 
)
] , \linebreak [0] 
[
\ast 
*_{f}
\ast 
] , \linebreak [0] 
[
\ast 
**
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
+ 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_1} 
\ast 
] , \linebreak [0] 
[
\ast 
- 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_1} 
\ast 
] , \linebreak [0] 
[(
\ast 
+
\ast 
)
] , \linebreak [0] 
[(
\ast 
-
\ast 
)
] , \linebreak [0] 
[
\ast 
+_{f}
\ast 
] , \linebreak [0] 
[
\ast 
-_{f}
\ast 
] , \linebreak [0] 
[
\ast 
++
\ast 
] , \linebreak [0] 
[R(
\ast 
) -- R(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 \mathrel{\in} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[|
\ast 
|
] , \linebreak [0] 
[if(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[Max(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Max(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 = 
\ast 
] , \linebreak [0] 
[
\ast 
 \neq 
\ast 
] , \linebreak [0] 
[
\ast 
 <= 
\ast 
] , \linebreak [0] 
[
\ast 
 < 
\ast 
] , \linebreak [0] 
[
\ast 
<_{f}
\ast 
] , \linebreak [0] 
[
\ast 
\leq_{f}
\ast 
] , \linebreak [0] 
[SF(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
 == 
\ast 
] , \linebreak [0] 
[
\ast 
 !!== 
\ast 
] , \linebreak [0] 
[
\ast 
 << 
\ast 
] , \linebreak [0] 
[
\ast 
 <<== 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cup \{ 
\ast 
\}
] , \linebreak [0] 
[
\ast 
\cup 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \{ 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel { \dot { . \, . } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { \dot { . \, . } } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { : \, : } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { {+} 2 \ast } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { : \, : }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { {+} 2 \ast }
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
,\linebreak [0] 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\stackrel {B}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {D}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {C}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {P}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\approx 
\ast 
] , \linebreak [0] 
[
\ast 
=
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {+}{\rightarrow }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t^*}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{r}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_t 
\ast 
] , \linebreak [0] 
[
\ast 
\subseteq_T 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{T}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{s}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in}^{\ast} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for}^{\ast} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_c 
\ast 
] , \linebreak [0] 
[
\ast 
< 
\ast 
] , \linebreak [0] 
[
\ast 
<' 
\ast 
] , \linebreak [0] 
[
\ast 
\le' 
\ast 
] , \linebreak [0] 
[
\ast 
= 
\ast 
] , \linebreak [0] 
[
\ast 
\neq 
\ast 
] , \linebreak [0] 
[
\ast 
{}^{var}
] , \linebreak [0] 
[
\ast 
\#^0
\ast 
] , \linebreak [0] 
[
\ast 
\#^1
\ast 
] , \linebreak [0] 
[
\ast 
\#^*
\ast 
] , \linebreak [0] 
[
\ast 
\!\mathrel{==}\! 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\subseteq} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
{\neg }
\ast 
] , \linebreak [0] 
[\dot{\neg}\,(
\ast 
)n
] , \linebreak [0] 
[
\ast 
 \mathrel{\notin} 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\neq} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\wedge 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\tilde{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\wedge_c 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\dot{\wedge}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\vee 
\ast 
] , \linebreak [0] 
[
\ast 
\parallel 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\vee}} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
 \mathrel{\dot{\vee}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\exists 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall_{obj} 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\exists 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\ddot{\Rightarrow}} 
\ast 
] , \linebreak [0] 
[
\ast 
\Rightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\Leftrightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\dot{\Leftrightarrow}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[ \{ ph \mathrel{\in} 
\ast 
\mid 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
:
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{spy}
\ast 
] , \linebreak [0] 
[
\ast 
!
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\left\{\protect \begin {array}{l}
\ast 
\\
\ast 
\protect \end {array}\right.
]
; 
\newline \mathbf {Preassociative} \newline 
[
\lambda 
\ast 
.
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
. 
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
] , \linebreak [0] 
[
{\bf if} \ 
\ast 
\ {\bf then} \ 
\ast 
\ {\bf else} \ 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
= 
\ast 
\mathbf{\ in\ } 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
\mathrel{\ddot{=}} 
\ast 
\mathrel{\ in\ } 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\#
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{} ^ { I }
] , \linebreak [0] 
[
\ast 
{} ^ { \rhd }
] , \linebreak [0] 
[
\ast 
{} ^ { V } 
] , \linebreak [0] 
[
\ast 
{} ^ { + } 
] , \linebreak [0] 
[
\ast 
{} ^ { - } 
] , \linebreak [0] 
[
\ast 
{} ^ { \ast } 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathop {\char64} 
\ast 
] , \linebreak [0] 
[
\ast 
\rhd 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\ast 
] , \linebreak [0] 
[
\ast 
\gg 
\ast 
] , \linebreak [0] 
[
\ast 
\unrhd 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\vdash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {i.e.} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\Pi 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel {\oplus} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
; 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\ proves\ 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbf{\ proof\ of\ } 
\ast 
: 
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: 
\ast 
\gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Last\ line \, 
\ast 
\gg 
\ast 
\,\Box
] , \linebreak [0] 
[
Line \, 
\ast 
: Premise \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: \mbox{Side-condition} \gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Local \gg 
\ast 
= 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Begin \, 
\ast 
 ; 
\ast 
 : End ; 
\ast 
] , \linebreak [0] 
[
Last\ block\ line \, 
\ast 
\gg 
\ast 
\,;
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{|} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\mathrm{,}}
\ast 
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\& 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\backslash \backslash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{linebreak[4]} 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \backslash 
\ast 
]
; 
]$
\end {flushleft}

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

\begin{flushleft}
$
[UniqueMember
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}uniqueMember
\mbox{''}
]
\linebreak[4] 
[UniqueMember(Type)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}uniqueMember(Type)
\mbox{''}
]
\linebreak[4] 
[SameSeries
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameSeries
\mbox{''}
]
\linebreak[4] 
[A4
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}a4
\mbox{''}
]
\linebreak[4] 
[SameMember
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameMember
\mbox{''}
]
\linebreak[4] 
[Qclosed(Addition)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}Qclosed(Addition)
\mbox{''}
]
\linebreak[4] 
[Qclosed(Multiplication)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}Qclosed(Multiplication)
\mbox{''}
]
\linebreak[4] 
[FromCartProd(1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}fromCartProd(1)
\mbox{''}
]
\linebreak[4] 
[
\mbox{1rule fromCartProd(2)}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}fromCartProd(2)
\mbox{''}
]
\linebreak[4] 
[constantRationalSeries(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}constantRationalSeries(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[cartProd(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}cartProd(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[Power(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}P(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[binaryUnion(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}binaryUnion(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[SetOfRationalSeries
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}setOfRationalSeries
\mbox{''}
]
\linebreak[4] 
[IsSubset(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}isSubset(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[(p
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}(p
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[(s
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}(s
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[(\cdots{})
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}cdots
\mbox{''}
]
\linebreak[4] 
[\texttt{Objekt-var}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\mbox{-}var
\mbox{''}
]
\linebreak[4] 
[\texttt{Ex-var}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex
\mbox{-}var
\mbox{''}
]
\linebreak[4] 
[\texttt{Ph-var}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}var
\mbox{''}
]
\linebreak[4] 
[\texttt{V\ae{}rdi}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}vaerdi
\mbox{''}
]
\linebreak[4] 
[\texttt{Variabel}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}variabel
\mbox{''}
]
\linebreak[4] 
[Op(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}op
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}op
\mbox{''}
]
\linebreak[4] 
[Op(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}op2
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}op2
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel {\ddot {==}} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}define
\mbox{-}equal
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}equal
\mbox{''}
]
\linebreak[4] 
[ContainsEmpty(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}contains
\mbox{-}empty
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}empty
\mbox{''}
]
\linebreak[4] 
[Nat(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}Nat(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[
Dedu(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_0(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[Dedu_{s}(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}side
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_1(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}one
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_2(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}four
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}four
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_5(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}five
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}condition
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}six
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}exception
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}six
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}conclude
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}exception
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_7(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}seven
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_8(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}eight
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[
Dedu_8^*(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1deduction
\linebreak [0]\ \hskip0em plus2.0em{}eight
\linebreak [0]\ \hskip0em plus2.0em{}star
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}1deduction
\mbox{''}
]
\linebreak[4] 
[Ex_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex1
\mbox{''}
]
\linebreak[4] 
[Ex_{2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex2
\mbox{''}
]
\linebreak[4] 
[Ex3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex3
\mbox{''}
]
\linebreak[4] 
[Ex_{10}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex10
\mbox{''}
]
\linebreak[4] 
[Ex_{20}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ex20
\mbox{''}
]
\linebreak[4] 
[
\ast 
 _{Ex}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}existential
\linebreak [0]\ \hskip0em plus2.0em{}var
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}var
\mbox{''}
]
\linebreak[4] 
[
\ast 
 ^{Ex}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}existential
\linebreak [0]\ \hskip0em plus2.0em{}var
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\mbox{-}sub
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\mbox{-}sub0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\mbox{-}sub1
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist
\mbox{-}sub
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[ph_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph1
\mbox{''}
]
\linebreak[4] 
[ph_{2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph2
\mbox{''}
]
\linebreak[4] 
[ph_{3}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph3
\mbox{''}
]
\linebreak[4] 
[
\ast 
 _{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}placeholder
\mbox{-}var
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}var
\mbox{''}
]
\linebreak[4] 
[
\ast 
 ^{Ph}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}placeholder
\mbox{-}var
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}sub
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}sub0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}sub1
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph
\mbox{-}sub
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\mbox{-}sub
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\mbox{-}sub1
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\mbox{-}sub
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}where
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}sub
\mbox{''}
]
\linebreak[4] 
[\mathsf {bs}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}big
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[ \mathsf {OBS}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}big
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[{\cal BS}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}big
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[\mathrm{\O}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[SystemQ
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}system
\linebreak [0]\ \hskip0em plus2.0em{}Q
\mbox{''}
]
\linebreak[4] 
[MP
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}mp
\mbox{''}
]
\linebreak[4] 
[Gen
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}gen
\mbox{''}
]
\linebreak[4] 
[Repetition
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}repetition
\mbox{''}
]
\linebreak[4] 
[Neg
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}ad
\linebreak [0]\ \hskip0em plus2.0em{}absurdum
\mbox{''}
]
\linebreak[4] 
[Ded
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}deduction
\mbox{''}
]
\linebreak[4] 
[ExistIntro
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}exist
\linebreak [0]\ \hskip0em plus2.0em{}intro
\mbox{''}
]
\linebreak[4] 
[Extensionality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}extensionality
\mbox{''}
]
\linebreak[4] 
[\O{}def
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[PairDef
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}definition
\mbox{''}
]
\linebreak[4] 
[UnionDef
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}union
\linebreak [0]\ \hskip0em plus2.0em{}definition
\mbox{''}
]
\linebreak[4] 
[PowerDef
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}definition
\mbox{''}
]
\linebreak[4] 
[SeparationDef
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}separation
\linebreak [0]\ \hskip0em plus2.0em{}definition
\mbox{''}
]
\linebreak[4] 
[AddDoubleNeg
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}add
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}neg
\mbox{''}
]
\linebreak[4] 
[RemoveDoubleNeg
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}remove
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}neg
\mbox{''}
]
\linebreak[4] 
[AndCommutativity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}and
\linebreak [0]\ \hskip0em plus2.0em{}commutativity
\mbox{''}
]
\linebreak[4] 
[AutoImply
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}auto
\linebreak [0]\ \hskip0em plus2.0em{}imply
\mbox{''}
]
\linebreak[4] 
[Contrapositive
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}contrapositive
\mbox{''}
]
\linebreak[4] 
[FirstConjunct
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}first
\linebreak [0]\ \hskip0em plus2.0em{}conjunct
\mbox{''}
]
\linebreak[4] 
[SecondConjunct
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}conjunct
\mbox{''}
]
\linebreak[4] 
[FromContradiction
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}contradiction
\mbox{''}
]
\linebreak[4] 
[FromDisjuncts
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}disjuncts
\mbox{''}
]
\linebreak[4] 
[IffCommutativity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}iff
\linebreak [0]\ \hskip0em plus2.0em{}commutativity
\mbox{''}
]
\linebreak[4] 
[IffFirst
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}iff
\linebreak [0]\ \hskip0em plus2.0em{}first
\mbox{''}
]
\linebreak[4] 
[IffSecond
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}iff
\linebreak [0]\ \hskip0em plus2.0em{}second
\mbox{''}
]
\linebreak[4] 
[ImplyTransitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}imply
\linebreak [0]\ \hskip0em plus2.0em{}transitivity
\mbox{''}
]
\linebreak[4] 
[JoinConjuncts
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}join
\linebreak [0]\ \hskip0em plus2.0em{}conjuncts
\mbox{''}
]
\linebreak[4] 
[MP2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mp2
\mbox{''}
]
\linebreak[4] 
[MP3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mp3
\mbox{''}
]
\linebreak[4] 
[MP4
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mp4
\mbox{''}
]
\linebreak[4] 
[MP5
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mp5
\mbox{''}
]
\linebreak[4] 
[MT
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}mt
\mbox{''}
]
\linebreak[4] 
[NegativeMT
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negative
\linebreak [0]\ \hskip0em plus2.0em{}mt
\mbox{''}
]
\linebreak[4] 
[Technicality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}technicality
\mbox{''}
]
\linebreak[4] 
[Weakening
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}weakening
\mbox{''}
]
\linebreak[4] 
[WeakenOr1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}weaken
\linebreak [0]\ \hskip0em plus2.0em{}or
\linebreak [0]\ \hskip0em plus2.0em{}first
\mbox{''}
]
\linebreak[4] 
[WeakenOr2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}weaken
\linebreak [0]\ \hskip0em plus2.0em{}or
\linebreak [0]\ \hskip0em plus2.0em{}second
\mbox{''}
]
\linebreak[4] 
[Formula2Pair
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}formula2pair
\mbox{''}
]
\linebreak[4] 
[Pair2Formula
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}pair2formula
\mbox{''}
]
\linebreak[4] 
[Formula2Union
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}formula2union
\mbox{''}
]
\linebreak[4] 
[Union2Formula
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}union2formula
\mbox{''}
]
\linebreak[4] 
[Formula2Sep
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}formula2separation
\mbox{''}
]
\linebreak[4] 
[Sep2Formula
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}separation2formula
\mbox{''}
]
\linebreak[4] 
[Formula2Power
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}formula2power
\mbox{''}
]
\linebreak[4] 
[SubsetInPower
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subset
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[HelperPowerIsSub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset0
\mbox{''}
]
\linebreak[4] 
[PowerIsSub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[(Switch)HelperPowerIsSub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset0
\mbox{-}switch
\mbox{''}
]
\linebreak[4] 
[(Switch)PowerIsSub
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}power
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{-}switch
\mbox{''}
]
\linebreak[4] 
[ToSetEquality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}suff
\linebreak [0]\ \hskip0em plus2.0em{}condition
\mbox{''}
]
\linebreak[4] 
[HelperToSetEquality(t)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}suff
\linebreak [0]\ \hskip0em plus2.0em{}condition(t)0
\mbox{''}
]
\linebreak[4] 
[ToSetEquality(t)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}suff
\linebreak [0]\ \hskip0em plus2.0em{}condition(t)
\mbox{''}
]
\linebreak[4] 
[HelperFromSetEquality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}skip
\linebreak [0]\ \hskip0em plus2.0em{}quantifier
\mbox{''}
]
\linebreak[4] 
[FromSetEquality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition
\mbox{''}
]
\linebreak[4] 
[HelperReflexivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}reflexivity0
\mbox{''}
]
\linebreak[4] 
[Reflexivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}reflexivity
\mbox{''}
]
\linebreak[4] 
[HelperSymmetry
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}symmetry0
\mbox{''}
]
\linebreak[4] 
[Symmetry
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}symmetry
\mbox{''}
]
\linebreak[4] 
[HelperTransitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}transitivity0
\mbox{''}
]
\linebreak[4] 
[Transitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}transitivity
\mbox{''}
]
\linebreak[4] 
[ERisReflexive
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}er
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}reflexive
\mbox{''}
]
\linebreak[4] 
[ERisSymmetric
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}er
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}symmetric
\mbox{''}
]
\linebreak[4] 
[ERisTransitive
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}er
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}transitive
\mbox{''}
]
\linebreak[4] 
[\O{}isSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[HelperMemberNot\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}member
\linebreak [0]\ \hskip0em plus2.0em{}not
\linebreak [0]\ \hskip0em plus2.0em{}empty0
\mbox{''}
]
\linebreak[4] 
[MemberNot\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}member
\linebreak [0]\ \hskip0em plus2.0em{}not
\linebreak [0]\ \hskip0em plus2.0em{}empty
\mbox{''}
]
\linebreak[4] 
[HelperUnique\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}unique
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set0
\mbox{''}
]
\linebreak[4] 
[Unique\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}unique
\linebreak [0]\ \hskip0em plus2.0em{}empty
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[==\!{}Reflexivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
{=}Reflexivity
\mbox{''}
]
\linebreak[4] 
[==\!{}Symmetry
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
{=}Symmetry
\mbox{''}
]
\linebreak[4] 
[Helper\!{}==\!{}Transitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
{=}Transitivity0
\mbox{''}
]
\linebreak[4] 
[\!{}==\!{}Transitivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
{=}Transitivity
\mbox{''}
]
\linebreak[4] 
[HelperTransferNotEq
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}transfer
\linebreak [0]\ \hskip0em plus2.0em{}
\char126 is0
\mbox{''}
]
\linebreak[4] 
[TransferNotEq
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}transfer
\linebreak [0]\ \hskip0em plus2.0em{}
\char126 is
\mbox{''}
]
\linebreak[4] 
[HelperPairSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}subset0
\mbox{''}
]
\linebreak[4] 
[Helper(2)PairSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}subset1
\mbox{''}
]
\linebreak[4] 
[PairSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[SamePair
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}pair
\mbox{''}
]
\linebreak[4] 
[SameSingleton
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}singleton
\mbox{''}
]
\linebreak[4] 
[UnionSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}union
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[SameUnion
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}union
\mbox{''}
]
\linebreak[4] 
[SeparationSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}separation
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[SameSeparation
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}separation
\mbox{''}
]
\linebreak[4] 
[SameBinaryUnion
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}binary
\linebreak [0]\ \hskip0em plus2.0em{}union
\mbox{''}
]
\linebreak[4] 
[IntersectionSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intersection
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[SameIntersection
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}same
\linebreak [0]\ \hskip0em plus2.0em{}intersection
\mbox{''}
]
\linebreak[4] 
[AutoMember
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}auto
\linebreak [0]\ \hskip0em plus2.0em{}member
\mbox{''}
]
\linebreak[4] 
[HelperEqSysNot\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eq
\mbox{-}system
\linebreak [0]\ \hskip0em plus2.0em{}not
\linebreak [0]\ \hskip0em plus2.0em{}empty0
\mbox{''}
]
\linebreak[4] 
[EqSysNot\O{}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eq
\mbox{-}system
\linebreak [0]\ \hskip0em plus2.0em{}not
\linebreak [0]\ \hskip0em plus2.0em{}empty
\mbox{''}
]
\linebreak[4] 
[HelperEqSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eq
\linebreak [0]\ \hskip0em plus2.0em{}subset0
\mbox{''}
]
\linebreak[4] 
[EqSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eq
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[HelperEqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition0
\mbox{''}
]
\linebreak[4] 
[EqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition
\mbox{''}
]
\linebreak[4] 
[HelperNoneEqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}none
\mbox{-}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition0
\mbox{''}
]
\linebreak[4] 
[Helper(2)NoneEqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}none
\mbox{-}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition1
\mbox{''}
]
\linebreak[4] 
[NoneEqNecessary
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}none
\mbox{-}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition
\mbox{''}
]
\linebreak[4] 
[EqClassIsSubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}class
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\mbox{''}
]
\linebreak[4] 
[EqClassesAreDisjoint
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}classes
\linebreak [0]\ \hskip0em plus2.0em{}are
\linebreak [0]\ \hskip0em plus2.0em{}disjoint
\mbox{''}
]
\linebreak[4] 
[AllDisjoint
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}all
\linebreak [0]\ \hskip0em plus2.0em{}disjoint
\mbox{''}
]
\linebreak[4] 
[AllDisjointImply
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}all
\linebreak [0]\ \hskip0em plus2.0em{}disjoint
\mbox{-}imply
\mbox{''}
]
\linebreak[4] 
[BSsubset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}bs
\linebreak [0]\ \hskip0em plus2.0em{}subset
\linebreak [0]\ \hskip0em plus2.0em{}union(bs/r)
\mbox{''}
]
\linebreak[4] 
[Union(BS/R)subset
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}union(bs/r)
\linebreak [0]\ \hskip0em plus2.0em{}subset
\linebreak [0]\ \hskip0em plus2.0em{}bs
\mbox{''}
]
\linebreak[4] 
[UnionIdentity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}union(bs/r)
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}bs
\mbox{''}
]
\linebreak[4] 
[EqSysIsPartition
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}theorem
\linebreak [0]\ \hskip0em plus2.0em{}eq
\mbox{-}system
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}partition
\mbox{''}
]
\linebreak[4] 
[(x1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}x1
\mbox{''}
]
\linebreak[4] 
[(x2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}x2
\mbox{''}
]
\linebreak[4] 
[(y1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}y1
\mbox{''}
]
\linebreak[4] 
[(y2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}y2
\mbox{''}
]
\linebreak[4] 
[(v1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}v1
\mbox{''}
]
\linebreak[4] 
[(v2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}v2
\mbox{''}
]
\linebreak[4] 
[(v3)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}v3
\mbox{''}
]
\linebreak[4] 
[(v4)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}v4
\mbox{''}
]
\linebreak[4] 
[(v2n)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}v2n
\mbox{''}
]
\linebreak[4] 
[(m1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}m1
\mbox{''}
]
\linebreak[4] 
[(m2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}m2
\mbox{''}
]
\linebreak[4] 
[(n1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}n1
\mbox{''}
]
\linebreak[4] 
[(n2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}n2
\mbox{''}
]
\linebreak[4] 
[(n3)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}n3
\mbox{''}
]
\linebreak[4] 
[(\epsilon)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}ep
\mbox{''}
]
\linebreak[4] 
[(\epsilon)_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}ep1
\mbox{''}
]
\linebreak[4] 
[(\epsilon 2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}ep2
\mbox{''}
]
\linebreak[4] 
[(fep)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fep
\mbox{''}
]
\linebreak[4] 
[(fx)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fx
\mbox{''}
]
\linebreak[4] 
[(fy)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fy
\mbox{''}
]
\linebreak[4] 
[(fz)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fz
\mbox{''}
]
\linebreak[4] 
[(fu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fu
\mbox{''}
]
\linebreak[4] 
[(fv)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fv
\mbox{''}
]
\linebreak[4] 
[(fw)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fw
\mbox{''}
]
\linebreak[4] 
[(rx)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}rx
\mbox{''}
]
\linebreak[4] 
[(ry)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}ry
\mbox{''}
]
\linebreak[4] 
[(rz)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}rz
\mbox{''}
]
\linebreak[4] 
[(ru)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}ru
\mbox{''}
]
\linebreak[4] 
[(sx)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}sx
\mbox{''}
]
\linebreak[4] 
[(sx1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}sx1
\mbox{''}
]
\linebreak[4] 
[(sy)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}sy
\mbox{''}
]
\linebreak[4] 
[(sy1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}sy1
\mbox{''}
]
\linebreak[4] 
[(sz)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}sz
\mbox{''}
]
\linebreak[4] 
[(sz1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}sz1
\mbox{''}
]
\linebreak[4] 
[(su)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}su
\mbox{''}
]
\linebreak[4] 
[(su1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}su1
\mbox{''}
]
\linebreak[4] 
[(fxs)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fxs
\mbox{''}
]
\linebreak[4] 
[(fys)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}fys
\mbox{''}
]
\linebreak[4] 
[(crs1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}crs1
\mbox{''}
]
\linebreak[4] 
[(f1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}f1
\mbox{''}
]
\linebreak[4] 
[(f2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}f2
\mbox{''}
]
\linebreak[4] 
[(f3)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}f3
\mbox{''}
]
\linebreak[4] 
[(f4)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}f4
\mbox{''}
]
\linebreak[4] 
[(op1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}op1
\mbox{''}
]
\linebreak[4] 
[(op2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}op2
\mbox{''}
]
\linebreak[4] 
[(r1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}r1
\mbox{''}
]
\linebreak[4] 
[(s1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}s1
\mbox{''}
]
\linebreak[4] 
[(s2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}var
\linebreak [0]\ \hskip0em plus2.0em{}s2
\mbox{''}
]
\linebreak[4] 
[X_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}x1
\mbox{''}
]
\linebreak[4] 
[X_{2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}x2
\mbox{''}
]
\linebreak[4] 
[Y_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}y1
\mbox{''}
]
\linebreak[4] 
[Y_{2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}y2
\mbox{''}
]
\linebreak[4] 
[V_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}v1
\mbox{''}
]
\linebreak[4] 
[V_{2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}v2
\mbox{''}
]
\linebreak[4] 
[V_{3}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}v3
\mbox{''}
]
\linebreak[4] 
[V_{4}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}v4
\mbox{''}
]
\linebreak[4] 
[V_{2n}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}v2n
\mbox{''}
]
\linebreak[4] 
[M_{1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}m1
\mbox{''}
]
\linebreak[4] 
[M_{2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}m2
\mbox{''}
]
\linebreak[4] 
[N_{1} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}n1
\mbox{''}
]
\linebreak[4] 
[N_{2} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}n2
\mbox{''}
]
\linebreak[4] 
[N_{3} 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}n3
\mbox{''}
]
\linebreak[4] 
[\epsilon
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}ep
\mbox{''}
]
\linebreak[4] 
[\epsilon 1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}ep1
\mbox{''}
]
\linebreak[4] 
[\epsilon 2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}ep2
\mbox{''}
]
\linebreak[4] 
[FX
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fx
\mbox{''}
]
\linebreak[4] 
[FY
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fy
\mbox{''}
]
\linebreak[4] 
[FZ
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fz
\mbox{''}
]
\linebreak[4] 
[FU
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fu
\mbox{''}
]
\linebreak[4] 
[FV
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fv
\mbox{''}
]
\linebreak[4] 
[FW
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fw
\mbox{''}
]
\linebreak[4] 
[FEP
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fep
\mbox{''}
]
\linebreak[4] 
[RX
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}rx
\mbox{''}
]
\linebreak[4] 
[RY
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}ry
\mbox{''}
]
\linebreak[4] 
[RZ
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}rz
\mbox{''}
]
\linebreak[4] 
[RU
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}ru
\mbox{''}
]
\linebreak[4] 
[(SX)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}sx
\mbox{''}
]
\linebreak[4] 
[(SX1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}sx1
\mbox{''}
]
\linebreak[4] 
[(SY)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}sy
\mbox{''}
]
\linebreak[4] 
[(SY1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}sy1
\mbox{''}
]
\linebreak[4] 
[(SZ)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}sz
\mbox{''}
]
\linebreak[4] 
[(SZ1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}sz1
\mbox{''}
]
\linebreak[4] 
[(SU)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}su
\mbox{''}
]
\linebreak[4] 
[(SU1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}su1
\mbox{''}
]
\linebreak[4] 
[FXS
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fxs
\mbox{''}
]
\linebreak[4] 
[FYS
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}fys
\mbox{''}
]
\linebreak[4] 
[(F1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}f1
\mbox{''}
]
\linebreak[4] 
[(F2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}f2
\mbox{''}
]
\linebreak[4] 
[(F3)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}f3
\mbox{''}
]
\linebreak[4] 
[(F4)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}f4
\mbox{''}
]
\linebreak[4] 
[(OP1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}op1
\mbox{''}
]
\linebreak[4] 
[(OP2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}op2
\mbox{''}
]
\linebreak[4] 
[(R1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}r1
\mbox{''}
]
\linebreak[4] 
[(S1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}s1
\mbox{''}
]
\linebreak[4] 
[(S2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}meta
\linebreak [0]\ \hskip0em plus2.0em{}s2
\mbox{''}
]
\linebreak[4] 
[(EPob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}ep
\mbox{''}
]
\linebreak[4] 
[(CRS1ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}crs1
\mbox{''}
]
\linebreak[4] 
[(F1ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}f1
\mbox{''}
]
\linebreak[4] 
[(F2ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}f2
\mbox{''}
]
\linebreak[4] 
[(F3ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}f3
\mbox{''}
]
\linebreak[4] 
[(F4ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}f4
\mbox{''}
]
\linebreak[4] 
[(N1ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}n1
\mbox{''}
]
\linebreak[4] 
[(N2ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}n2
\mbox{''}
]
\linebreak[4] 
[(OP1ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}op1
\mbox{''}
]
\linebreak[4] 
[(OP2ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}op2
\mbox{''}
]
\linebreak[4] 
[(R1ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}r1
\mbox{''}
]
\linebreak[4] 
[(S1ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}s1
\mbox{''}
]
\linebreak[4] 
[(S2ob)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}object
\linebreak [0]\ \hskip0em plus2.0em{}s2
\mbox{''}
]
\linebreak[4] 
[ph_{4}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph4
\mbox{''}
]
\linebreak[4] 
[ph_{5}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph5
\mbox{''}
]
\linebreak[4] 
[ph_{6}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ph6
\mbox{''}
]
\linebreak[4] 
[NAT
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}NAT
\mbox{''}
]
\linebreak[4] 
[RATIONAL_SERIES
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}RATIONAL
\_SERIES
\mbox{''}
]
\linebreak[4] 
[SERIES
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}SERIES
\mbox{''}
]
\linebreak[4] 
[SetOfReals
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}setOfReals
\mbox{''}
]
\linebreak[4] 
[SetOfFxs
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}setOfFxs
\mbox{''}
]
\linebreak[4] 
[N
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}N
\mbox{''}
]
\linebreak[4] 
[Q
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}Q
\mbox{''}
]
\linebreak[4] 
[X
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}X
\mbox{''}
]
\linebreak[4] 
[xs
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}xs
\mbox{''}
]
\linebreak[4] 
[xaF
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}xsF
\mbox{''}
]
\linebreak[4] 
[ysF
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}ysF
\mbox{''}
]
\linebreak[4] 
[us
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}us
\mbox{''}
]
\linebreak[4] 
[usFoelge
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}usF
\mbox{''}
]
\linebreak[4] 
[0
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}0
\mbox{''}
]
\linebreak[4] 
[1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1
\mbox{''}
]
\linebreak[4] 
[(-1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}(
\mbox{-}1)
\mbox{''}
]
\linebreak[4] 
[2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}2
\mbox{''}
]
\linebreak[4] 
[3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}3
\mbox{''}
]
\linebreak[4] 
[1/2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1/2
\mbox{''}
]
\linebreak[4] 
[1/3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1/3
\mbox{''}
]
\linebreak[4] 
[2/3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}2/3
\mbox{''}
]
\linebreak[4] 
[0f
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}0f
\mbox{''}
]
\linebreak[4] 
[
\mbox{1f}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1f
\mbox{''}
]
\linebreak[4] 
[00
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}00
\mbox{''}
]
\linebreak[4] 
[
\mbox{01}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}01
\mbox{''}
]
\linebreak[4] 
[(--01)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}(
\mbox{-}
\mbox{-}01)
\mbox{''}
]
\linebreak[4] 
[02
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}02
\mbox{''}
]
\linebreak[4] 
[01//02
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}01//02
\mbox{''}
]
\linebreak[4] 
[PlusAssociativity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}plusAssociativity(R)
\mbox{''}
]
\linebreak[4] 
[PlusAssociativity(R)XX
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}plusAssociativity(R)XX
\mbox{''}
]
\linebreak[4] 
[Plus0(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}plus0(R)
\mbox{''}
]
\linebreak[4] 
[Negative(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negative(R)
\mbox{''}
]
\linebreak[4] 
[Times1(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}times1(R)
\mbox{''}
]
\linebreak[4] 
[lessAddition(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessAddition(R)
\mbox{''}
]
\linebreak[4] 
[PlusCommutativity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}plusCommutativity(R)
\mbox{''}
]
\linebreak[4] 
[LeqAntisymmetry(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqAntisymmetry(R)
\mbox{''}
]
\linebreak[4] 
[LeqTransitivity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqTransitivity(R)
\mbox{''}
]
\linebreak[4] 
[leqAddition(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqAddition(R)
\mbox{''}
]
\linebreak[4] 
[Distribution(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}distribution(R)
\mbox{''}
]
\linebreak[4] 
[A4(Axiom)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}a4
\mbox{''}
]
\linebreak[4] 
[InductionAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}induction
\mbox{''}
]
\linebreak[4] 
[EqualityAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}equality
\mbox{''}
]
\linebreak[4] 
[EqLeqAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}eqLeq
\mbox{''}
]
\linebreak[4] 
[EqAdditionAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}eqAddition
\mbox{''}
]
\linebreak[4] 
[EqMultiplicationAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}eqMultiplication
\mbox{''}
]
\linebreak[4] 
[QisClosed(Reciprocal)(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}QisClosed(reciprocal)
\mbox{''}
]
\linebreak[4] 
[QisClosed(Reciprocal)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}QisClosed(reciprocal)
\mbox{''}
]
\linebreak[4] 
[QisClosed(Negative)(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}QisClosed(negative)
\mbox{''}
]
\linebreak[4] 
[QisClosed(Negative)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}QisClosed(negative)
\mbox{''}
]
\linebreak[4] 
[leqReflexivity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}leqReflexivity
\mbox{''}
]
\linebreak[4] 
[leqAntisymmetryAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}leqAntisymmetry
\mbox{''}
]
\linebreak[4] 
[leqTransitivityAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}leqTransitivity
\mbox{''}
]
\linebreak[4] 
[leqTotality
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}leqTotality
\mbox{''}
]
\linebreak[4] 
[leqAdditionAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}leqAddition
\mbox{''}
]
\linebreak[4] 
[leqMultiplicationAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}leqMultiplication
\mbox{''}
]
\linebreak[4] 
[plusAssociativity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}plusAssociativity
\mbox{''}
]
\linebreak[4] 
[plusCommutativity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}plusCommutativity
\mbox{''}
]
\linebreak[4] 
[Negative
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}negative
\mbox{''}
]
\linebreak[4] 
[plus0
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}plus0
\mbox{''}
]
\linebreak[4] 
[timesAssociativity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}timesAssociativity
\mbox{''}
]
\linebreak[4] 
[timesCommutativity
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}timesCommutativity
\mbox{''}
]
\linebreak[4] 
[ReciprocalAxiom
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}reciprocal
\mbox{''}
]
\linebreak[4] 
[times1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}times1
\mbox{''}
]
\linebreak[4] 
[Distribution
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}distribution
\mbox{''}
]
\linebreak[4] 
[0not1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}0not1
\mbox{''}
]
\linebreak[4] 
[
\mbox{lemma eqLeq(R)}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eqLeq(R)
\mbox{''}
]
\linebreak[4] 
[TimesAssociativity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}timesAssociativity(R)
\mbox{''}
]
\linebreak[4] 
[TimesCommutativity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}timesCommutativity(R)
\mbox{''}
]
\linebreak[4] 
[(Adgic)SameR
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}adhoc
\linebreak [0]\ \hskip0em plus2.0em{}sameR
\mbox{''}
]
\linebreak[4] 
[Separation2formula(1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}separation2formula(1)
\mbox{''}
]
\linebreak[4] 
[Separation2formula(2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}separation2formula(2)
\mbox{''}
]
\linebreak[4] 
[Cauchy
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}cauchy
\mbox{''}
]
\linebreak[4] 
[PlusF
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}plusF
\mbox{''}
]
\linebreak[4] 
[ReciprocalF
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}reciprocalF
\mbox{''}
]
\linebreak[4] 
[From==
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}from
{=}
{=}
\mbox{''}
]
\linebreak[4] 
[To==
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}to
{=}
{=}
\mbox{''}
]
\linebreak[4] 
[FromInR
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}fromInR
\mbox{''}
]
\linebreak[4] 
[PlusR(Sym)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}plusR(Sym)
\mbox{''}
]
\linebreak[4] 
[ReciprocalR(Axiom)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}reciprocalR
\mbox{''}
]
\linebreak[4] 
[LessMinus1(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}lessMinus1(N)
\mbox{''}
]
\linebreak[4] 
[Nonnegative(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}nonnegative(N)
\mbox{''}
]
\linebreak[4] 
[US0
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}US0
\mbox{''}
]
\linebreak[4] 
[NextXS(UpperBound)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}nextXS(upperBound)
\mbox{''}
]
\linebreak[4] 
[NextXS(NoUpperBound)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}nextXS(noUpperBound)
\mbox{''}
]
\linebreak[4] 
[NextUS(UpperBound)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}nextUS(upperBound)
\mbox{''}
]
\linebreak[4] 
[NextUS(NoUpperBound)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}nextUS(noUpperBound)
\mbox{''}
]
\linebreak[4] 
[ExpZero
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}expZero
\mbox{''}
]
\linebreak[4] 
[ExpPositive
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}expPositive
\mbox{''}
]
\linebreak[4] 
[ExpZero(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}expZero(R)
\mbox{''}
]
\linebreak[4] 
[ExpPositive(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}expPositive(R)
\mbox{''}
]
\linebreak[4] 
[BSzero
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}zero
\mbox{''}
]
\linebreak[4] 
[BSpositive
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}positive
\mbox{''}
]
\linebreak[4] 
[UStelescope(Zero)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}UStelescope
\linebreak [0]\ \hskip0em plus2.0em{}zero
\mbox{''}
]
\linebreak[4] 
[UStelescope(Positive)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}UStelescope
\linebreak [0]\ \hskip0em plus2.0em{}positive
\mbox{''}
]
\linebreak[4] 
[EqAddition(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}adhoc
\linebreak [0]\ \hskip0em plus2.0em{}eqAddition(R)
\mbox{''}
]
\linebreak[4] 
[FromLimit
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}fromLimit
\mbox{''}
]
\linebreak[4] 
[ToUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}toUpperBound
\mbox{''}
]
\linebreak[4] 
[FromUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}fromUpperBound
\mbox{''}
]
\linebreak[4] 
[USisUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}USisUpperBound
\mbox{''}
]
\linebreak[4] 
[0not1(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}0not1(R)
\mbox{''}
]
\linebreak[4] 
[ExpUnbounded(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}expUnbounded
\mbox{''}
]
\linebreak[4] 
[FromLeq(Advanced)(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}fromLeq(Advanced)(N)
\mbox{''}
]
\linebreak[4] 
[FromLeastUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}fromLeastUpperBound
\mbox{''}
]
\linebreak[4] 
[ToLeastUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}toLeastUpperBound
\mbox{''}
]
\linebreak[4] 
[XSisNotUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}XSisNotUpperBound
\mbox{''}
]
\linebreak[4] 
[ysFGreater
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}ysFGreater
\mbox{''}
]
\linebreak[4] 
[ysFLess
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}ysFLess
\mbox{''}
]
\linebreak[4] 
[SmallInverse
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1rule
\linebreak [0]\ \hskip0em plus2.0em{}smallInverse
\mbox{''}
]
\linebreak[4] 
[NatType
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}natType
\mbox{''}
]
\linebreak[4] 
[RationalType
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}rationalType
\mbox{''}
]
\linebreak[4] 
[SeriesType
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}seriesType
\mbox{''}
]
\linebreak[4] 
[Max
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}max
\mbox{''}
]
\linebreak[4] 
[Numerical
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}numerical
\mbox{''}
]
\linebreak[4] 
[NumericalF
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}numericalF
\mbox{''}
]
\linebreak[4] 
[MemberOfSeries(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}axiom
\linebreak [0]\ \hskip0em plus2.0em{}memberOfSeries
\mbox{''}
]
\linebreak[4] 
[JoinConjuncts(2conditions)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}doubly
\linebreak [0]\ \hskip0em plus2.0em{}conditioned
\linebreak [0]\ \hskip0em plus2.0em{}join
\linebreak [0]\ \hskip0em plus2.0em{}conjuncts
\mbox{''}
]
\linebreak[4] 
[
\mbox{prop lemma imply negation}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}imply
\linebreak [0]\ \hskip0em plus2.0em{}negation
\mbox{''}
]
\linebreak[4] 
[TND
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}tertium
\linebreak [0]\ \hskip0em plus2.0em{}non
\linebreak [0]\ \hskip0em plus2.0em{}datur
\mbox{''}
]
\linebreak[4] 
[FromNegatedImply
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}imply
\mbox{''}
]
\linebreak[4] 
[ToNegatedImply
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}to
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}imply
\mbox{''}
]
\linebreak[4] 
[FromNegated(2*Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}imply
\mbox{''}
]
\linebreak[4] 
[FromNegatedAnd
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}and
\mbox{''}
]
\linebreak[4] 
[FromNegatedOr
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}or
\mbox{''}
]
\linebreak[4] 
[ToNegatedOr
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}to
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}or
\mbox{''}
]
\linebreak[4] 
[FromNegations
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}negations
\mbox{''}
]
\linebreak[4] 
[From3Disjuncts
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}three
\linebreak [0]\ \hskip0em plus2.0em{}disjuncts
\mbox{''}
]
\linebreak[4] 
[From2*2Disjuncts
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}times
\linebreak [0]\ \hskip0em plus2.0em{}two
\linebreak [0]\ \hskip0em plus2.0em{}disjuncts
\mbox{''}
]
\linebreak[4] 
[NegateDisjunct1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negate
\linebreak [0]\ \hskip0em plus2.0em{}first
\linebreak [0]\ \hskip0em plus2.0em{}disjunct
\mbox{''}
]
\linebreak[4] 
[NegateDisjunct2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negate
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}disjunct
\mbox{''}
]
\linebreak[4] 
[ExpandDisjuncts
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expand
\linebreak [0]\ \hskip0em plus2.0em{}disjuncts
\mbox{''}
]
\linebreak[4] 
[SENC1
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition(1)
\mbox{''}
]
\linebreak[4] 
[SENC2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}equality
\linebreak [0]\ \hskip0em plus2.0em{}nec
\linebreak [0]\ \hskip0em plus2.0em{}condition(2)
\mbox{''}
]
\linebreak[4] 
[LessLeq(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessLeq(R)
\mbox{''}
]
\linebreak[4] 
[MemberOfSeries
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}memberOfSeries
\mbox{''}
]
\linebreak[4] 
[memberOfSeries(Type)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}memberOfSeries(Type)
\mbox{''}
]
\linebreak[4] 
[ 
\ast 
 (exp) 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{\char94}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[R(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}R(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[--R(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox{-}
\mbox{-}R(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[rec
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1/
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 / 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}eq
\mbox{-}system
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}modulo
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \cap 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}intersection
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}intersection
\mbox{''}
]
\linebreak[4] 
[
\ast 
[
\ast 
]
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}[
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{};
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}]
\mbox{''}
]
\linebreak[4] 
[\cup 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}union
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}union
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\cup} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}binary
\mbox{-}union
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}union
\mbox{''}
]
\linebreak[4] 
[P(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}power
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}power
\mbox{''}
]
\linebreak[4] 
[\{
\ast 
\}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}singleton
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}singleton
\mbox{''}
]
\linebreak[4] 
[StateExpand(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}stateExpand(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[extractSeries(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}extractSeries(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[SetOfSeries(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}setOfSeries(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[--Macro(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox{-}
\mbox{-}Macro(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[ExpandList(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}expandList(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[**Macro(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
{*}
{*}Macro(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[++Macro(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
{+}
{+}Macro(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[<<Macro(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
{<}
{<}Macro(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[||Macro(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}||Macro(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[01//Macro(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}01//Macro(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[UB(
\ast 
, 
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}upperBound(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[LUB(
\ast 
, 
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}leastUpperBound(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[BS(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}base(1/2)Sum(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[UStelescope(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}UStelescope(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[|f
\ast 
|
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}|f
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}|
\mbox{''}
]
\linebreak[4] 
[|r
\ast 
|
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}|r
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}|
\mbox{''}
]
\linebreak[4] 
[Limit(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}limit(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[Union(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}U(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[IsOrderedPair(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}isOrderedPair(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[IsRelation(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}isRelation(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[isFunction(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}isFunction(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[IsSeries(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}isSeries(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[IsNatural(
\ast 
,\ast
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}isNatural(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[OrderedPair(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}(o
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[TypeNat(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}typeNat(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[TypeNat0(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}typeNat0(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[TypeRational(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}typeRational(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[TypeRational0(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}typeRational0(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[TypeSeries(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}typeSeries(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[Typeseries0(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}typeSeries0(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[\{
\ast 
,
\ast 
\}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}pair
\mbox{''}
]
\linebreak[4] 
[\langle 
\ast 
,
\ast 
\rangle
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}ordered
\linebreak [0]\ \hskip0em plus2.0em{}pair
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}comma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}pair
\mbox{''}
]
\linebreak[4] 
[(-u
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox{-}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[-_{f}
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox{-}f
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[(--
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox{-}
\mbox{-}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[1f/
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}1f/
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[01//temp
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}01//
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}related
\linebreak [0]\ \hskip0em plus2.0em{}to
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}under
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[ReflRel(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}reflexive
\linebreak [0]\ \hskip0em plus2.0em{}relation
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[SymRel(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}symmetric
\linebreak [0]\ \hskip0em plus2.0em{}relation
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[TransRel(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}transitive
\linebreak [0]\ \hskip0em plus2.0em{}relation
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[EqRel(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}relation
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[[
\ast 
 \mathrel{\in} 
\ast 
]_{
\ast 
}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}equivalence
\linebreak [0]\ \hskip0em plus2.0em{}class
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}modulo
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[Partition(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}partition
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[(
\ast 
*
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
*_{f}
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{*}f
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
**
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{*}
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[(
\ast 
+
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{+}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[(
\ast 
-
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
+_{f}
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{+}f
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
-_{f}
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}f
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
++
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{+}
{+}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[R(
\ast 
) -- R(
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}R(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}
\mbox{-}
\linebreak [0]\ \hskip0em plus2.0em{}R(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\in} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}in0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[|
\ast 
|
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}|
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}|
\mbox{''}
]
\linebreak[4] 
[if(
\ast 
,
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}if(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[Max(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}max(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[Max(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}maxR(
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{},
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{})
\mbox{''}
]
\linebreak[4] 
[
\ast 
 = 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \neq 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}!
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 <= 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 < 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
<_{f}
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{<}f
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
\leq_{f}
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
{=}f
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[SF(
\ast 
,
\ast 
)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}sameF
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 == 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 !!== 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}!!
{=}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 << 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
{<}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 <<== 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
{<}
{=}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
\!\mathrel{==}\! 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\subseteq} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}is
\linebreak [0]\ \hskip0em plus2.0em{}subset
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[\dot{\neg}\,(
\ast 
)n
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}not0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\notin} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}
\char126 in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\neq} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}zermelo
\linebreak [0]\ \hskip0em plus2.0em{}
\char126 is
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
\mathrel{\dot{\wedge}} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}and0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
 \mathrel{\dot{\vee}} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}or0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\exists 
\ast 
\colon 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}exist0
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}indeed
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[
\ast 
\mathrel{\dot{\Leftrightarrow}} 
\ast 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}iff
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\mbox{''}
]
\linebreak[4] 
[ \{ ph \mathrel{\in} 
\ast 
\mid 
\ast 
\}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}the
\linebreak [0]\ \hskip0em plus2.0em{}set
\linebreak [0]\ \hskip0em plus2.0em{}of
\linebreak [0]\ \hskip0em plus2.0em{}ph
\linebreak [0]\ \hskip0em plus2.0em{}in
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}such
\linebreak [0]\ \hskip0em plus2.0em{}that
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {\tt \char34}
\linebreak [0]\ \hskip0em plus2.0em{}end
\linebreak [0]\ \hskip0em plus2.0em{}set
\mbox{''}
]
\linebreak[4] 
[kvanti
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}kvanti
\mbox{''}
]
\linebreak[4] $
\end{flushleft}

\newpage


\section{\TeX{} definitioner} \label{sec:tex}

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

\item $
[kvanti
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}kvanti
\mbox{''}
]$ 

\item $
[(\cdots{})
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox {$\backslash $}cdots
\{
\})
\mbox{''}
]$ 

\item $
[\texttt{Objekt-var}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}texttt
\{Objekt
\mbox{-}var
\}
\mbox{''}
]$ 

\item $
[\texttt{Ex-var}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}texttt
\{Ex
\mbox{-}var
\}
\mbox{''}
]$ 

\item $
[\texttt{Ph-var}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}texttt
\{Ph
\mbox{-}var
\}
\mbox{''}
]$ 

\item $
[\texttt{V\ae{}rdi}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}texttt
\{V
\mbox {$\backslash $}ae
\{
\}rdi
\}
\mbox{''}
]$ 

\item $
[\texttt{Variabel}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}texttt
\{Variabel
\}
\mbox{''}
]$

\item $
[Op(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Op(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[Op(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Op(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 \mathrel {\ddot {==}} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\linebreak [0]\ \hskip0em plus2.0em{}
\{
\mbox {$\backslash $}ddot
\linebreak [0]\ \hskip0em plus2.0em{}
\{
{=}
{=}
\}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[ContainsEmpty(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ContainsEmpty(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[
Dedu(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[
Dedu_0(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_0(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[Dedu_{s}(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Dedu
\_
\{s
\}(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

\item $
[
Dedu_1(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_1(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

\item $
[
Dedu_2(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_2(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

\item $
[
Dedu_3(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_3(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

\item $
[
Dedu_4(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_4(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

\item $
[
Dedu_4^*(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
,
\mathsf {u}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_4
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

\item $
[
Dedu_5(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_5(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$

\item $
[
Dedu_6(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_6(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

\item $
[
Dedu_6^*(
\mathsf {p}
,
\mathsf {c}
,
\mathsf {e}
,
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_6
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline ,
\#4.
\newline )
\mbox{''}
]$

\item $
[
Dedu_7(
\mathsf {p}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_7(
\#1.
\newline )
\mbox{''}
]$

\item $
[
Dedu_8(
\mathsf {p}
,
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_8(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[
Dedu_8^*(
\mathsf {p}
,
\mathsf {b}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline Dedu
\_8
{\char94}
{*}(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[Ex_{1}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Ex
\_
\{1
\}
\mbox{''}
]$ 

\item $
[Ex_{2}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Ex
\_
\{2
\}
\mbox{''}
]$ 

\item $
[Ex_{10}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Ex
\_
\{10
\}
\mbox{''}
]$ 

\item $
[Ex_{20}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Ex
\_
\{20
\}
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 _{Ex}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\_
\{Ex
\}
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 ^{Ex}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{\char94}
\{Ex
\}
\mbox{''}
]$ 

\item $
[\langle 
\mathsf{x}
{\equiv} 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Ex} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ex
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
{\equiv}^0 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Ex} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}0
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ex
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
{\equiv}^1 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Ex} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}1
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ex
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
{\equiv}^* 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Ex} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ex
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$


\item $
[ph_{1}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ph
\_
\{1
\}
\mbox{''}
]$ 

\item $
[ph_{2}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ph
\_
\{2
\}
\mbox{''}
]$ 

\item $
[ph_{3}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ph
\_
\{3
\}
\mbox{''}
]$ 

\item $
[ph_{4}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ph
\_
\{4
\}
\mbox{''}
]$ 

\item $
[ph_{5}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ph
\_
\{5
\}
\mbox{''}
]$ 

\item $
[ph_{6}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ph
\_
\{6
\}
\mbox{''}
]$

\item $
[
\ast 
 _{Ph} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 ^{Ph}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{\char94}
\{Ph
\}
\mbox{''}
]$ 

\item $
[\langle 
\mathsf{x}
{\equiv} 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Ph} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
{\equiv}^0 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Ph} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}0
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
{\equiv}^1 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Ph} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}1
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
{\equiv}^* 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Ph} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Ph
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\mathsf {bs}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}mathsf
\linebreak [0]\ \hskip0em plus2.0em{}
\{bs
\}
\mbox{''}
]$ 
\item $
[ \mathsf {OBS}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathsf
\linebreak [0]\ \hskip0em plus2.0em{}
\{OBS
\}
\mbox{''}
]$ 


\item $
[{\cal BS}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\{
\mbox {$\backslash $}cal
\linebreak [0]\ \hskip0em plus2.0em{}BS
\}
\mbox{''}
]$

\item $
[\mathrm{\O}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}mathrm
\{
\mbox {$\backslash $}O
\}
\mbox{''}
]$

\item $
[SystemQ
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SystemQ
\mbox{''}
]$

\item $
[MP
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MP
\mbox{''}
]$
\item $
[Gen
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Gen
\mbox{''}
]$
\item $
[Repetition
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Repetition
\mbox{''}
]$ 
\item $
[Neg
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Neg
\mbox{''}
]$
\item $
[Ded
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Ded
\mbox{''}
]$
\item $
[ExistIntro
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExistIntro
\mbox{''}
]$ 



\item $
[Extensionality
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Extensionality
\mbox{''}
]$
\item $
[\O{}def
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}O
\{
\}def
\mbox{''}
]$ 
\item $
[PairDef
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PairDef
\mbox{''}
]$ 
\item $
[UnionDef
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UnionDef
\mbox{''}
]$
\item $
[PowerDef
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PowerDef
\mbox{''}
]$ 
\item $
[SeparationDef
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SeparationDef
\mbox{''}
]$
\item $
[AddDoubleNeg
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}AddDoubleNeg
\mbox{''}
]$ 
\item $
[RemoveDoubleNeg
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}RemoveDoubleNeg
\mbox{''}
]$ 
\item $
[AndCommutativity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}AndCommutativity
\mbox{''}
]$ 
\item $
[AutoImply
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}AutoImply
\mbox{''}
]$ 
\item $
[Contrapositive
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Contrapositive
\mbox{''}
]$ 
\item $
[FirstConjunct
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FirstConjunct
\mbox{''}
]$ 
\item $
[SecondConjunct
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SecondConjunct
\mbox{''}
]$ 
\item $
[FromContradiction
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromContradiction
\mbox{''}
]$ 
\item $
[FromDisjuncts
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromDisjuncts
\mbox{''}
]$ 
\item $
[IffCommutativity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IffCommutativity
\mbox{''}
]$ 
\item $
[IffFirst
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IffFirst
\mbox{''}
]$ 

\item $
[IffSecond
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IffSecond
\mbox{''}
]$ 
\item $
[ImplyTransitivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ImplyTransitivity
\mbox{''}
]$ 



\item $
[JoinConjuncts
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}JoinConjuncts
\mbox{''}
]$ 

\item $
[MP2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MP2
\mbox{''}
]$ 

\item $
[MP3
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MP3
\mbox{''}
]$ 

\item $
[MP4
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MP4
\mbox{''}
]$ 

\item $
[MP5
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MP5
\mbox{''}
]$ 
\item $
[MT
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MT
\mbox{''}
]$ 

\item $
[NegativeMT
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeMT
\mbox{''}
]$ 
\item $
[Technicality
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Technicality
\mbox{''}
]$ 


\item $
[Weakening
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Weakening
\mbox{''}
]$ 

\item $
[WeakenOr1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}WeakenOr1
\mbox{''}
]$ 

\item $
[WeakenOr2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}WeakenOr2
\mbox{''}
]$ 

\item $
[Pair2Formula
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Pair2Formula
\mbox{''}
]$ 

\item $
[Formula2Pair
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Formula2Pair
\mbox{''}
]$ 

\item $
[Union2Formula
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Union2Formula
\mbox{''}
]$ 

\item $
[Formula2Union
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Formula2Union
\mbox{''}
]$ 

\item $
[Formula2Power
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Formula2Power
\mbox{''}
]$ 

\item $
[Sep2Formula
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Sep2Formula
\mbox{''}
]$ 

\item $
[Formula2Sep
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Formula2Sep
\mbox{''}
]$ 

\item $
[SubsetInPower
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubsetInPower
\mbox{''}
]$ 

\item $
[HelperPowerIsSub
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperPowerIsSub
\mbox{''}
]$ 

\item $
[PowerIsSub
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PowerIsSub
\mbox{''}
]$ 

\item $
[(Switch)HelperPowerIsSub
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(Switch)HelperPowerIsSub
\mbox{''}
]$ 

\item $
[(Switch)PowerIsSub
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(Switch)PowerIsSub
\mbox{''}
]$ 

\item $
[ToSetEquality
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToSetEquality
\mbox{''}
]$ 

\item $
[HelperToSetEquality(t)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperToSetEquality(t)
\mbox{''}
]$ 

\item $
[ToSetEquality(t)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToSetEquality(t)
\mbox{''}
]$ 

\item $
[HelperFromSetEquality
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperFromSetEquality
\mbox{''}
]$ 

\item $
[FromSetEquality
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromSetEquality
\mbox{''}
]$ 

\item $
[HelperReflexivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperReflexivity
\mbox{''}
]$ 

\item $
[Reflexivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Reflexivity
\mbox{''}
]$ 

\item $
[HelperSymmetry
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperSymmetry
\mbox{''}
]$ 

\item $
[Symmetry
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Symmetry
\mbox{''}
]$ 

\item $
[HelperTransitivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperTransitivity
\mbox{''}
]$

\item $
[Transitivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Transitivity
\mbox{''}
]$,

\item $
[ERisReflexive
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ERisReflexive
\mbox{''}
]$ 


\item $
[ERisSymmetric
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ERisSymmetric
\mbox{''}
]$ 

\item $
[ERisTransitive
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ERisTransitive
\mbox{''}
]$ 

\item $
[\O{}isSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}O
\{
\}isSubset
\mbox{''}
]$ 

\item $
[HelperMemberNot\O{}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperMemberNot
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[MemberNot\O{}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MemberNot
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[HelperUnique\O{}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperUnique
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[Unique\O{}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Unique
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[==\!{}Reflexivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
{=}
{=}
\mbox {$\backslash $}!
\{
\}Reflexivity
\mbox{''}
]$ 

\item $
[==\!{}Symmetry
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
{=}
{=}
\mbox {$\backslash $}!
\{
\}Symmetry
\mbox{''}
]$ 

\item $
[Helper\!{}==\!{}Transitivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Helper
\mbox {$\backslash $}!
\{
\}
{=}
{=}
\mbox {$\backslash $}!
\{
\}Transitivity
\mbox{''}
]$ 

\item $
[\!{}==\!{}Transitivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}!
\{
\}
{=}
{=}
\mbox {$\backslash $}!
\{
\}Transitivity
\mbox{''}
]$ 

\item $
[HelperTransferNotEq
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperTransferNotEq
\mbox{''}
]$ 

\item $
[TransferNotEq
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TransferNotEq
\mbox{''}
]$ 

\item $
[HelperPairSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperPairSubset
\mbox{''}
]$ 

\item $
[Helper(2)PairSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Helper(2)PairSubset
\mbox{''}
]$ 

\item $
[PairSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PairSubset
\mbox{''}
]$ 

\item $
[SamePair
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SamePair
\mbox{''}
]$ 

\item $
[SameSingleton
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameSingleton
\mbox{''}
]$ 


\item $
[UnionSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UnionSubset
\mbox{''}
]$ 

\item $
[SameUnion
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameUnion
\mbox{''}
]$ 



\item $
[SeparationSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SeparationSubset
\mbox{''}
]$

\item $
[SameSeparation
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameSeparation
\mbox{''}
]$ 

\item $
[SameBinaryUnion
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameBinaryUnion
\mbox{''}
]$ 

\item $
[IntersectionSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IntersectionSubset
\mbox{''}
]$ 

\item $
[SameIntersection
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameIntersection
\mbox{''}
]$ 

\item $
[AutoMember
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}AutoMember
\mbox{''}
]$ 

\item $
[HelperEqSysNot\O{}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperEqSysNot
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[EqSysNot\O{}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqSysNot
\mbox {$\backslash $}O
\{
\}
\mbox{''}
]$ 

\item $
[HelperEqSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperEqSubset
\mbox{''}
]$ 

\item $
[EqSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqSubset
\mbox{''}
]$ 

\item $
[EqNecessary
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqNecessary
\mbox{''}
]$ 

\item $
[HelperEqNecessary
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperEqNecessary
\mbox{''}
]$ 

\item $
[HelperNoneEqNecessary
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HelperNoneEqNecessary
\mbox{''}
]$ 

\item $
[Helper(2)NoneEqNecessary
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Helper(2)NoneEqNecessary
\mbox{''}
]$ 

\item $
[NoneEqNecessary
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NoneEqNecessary
\mbox{''}
]$ 

\item $
[EqClassIsSubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqClassIsSubset
\mbox{''}
]$ 

\item $
[EqClassesAreDisjoint
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqClassesAreDisjoint
\mbox{''}
]$ 


\item $
[AllDisjoint
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}AllDisjoint
\mbox{''}
]$ 

\item $
[AllDisjointImply
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}AllDisjointImply
\mbox{''}
]$ 

\item $
[BSsubset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BSsubset
\mbox{''}
]$ 

\item $
[Union(BS/R)subset
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Union(BS/R)subset
\mbox{''}
]$ 

\item $
[UnionIdentity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UnionIdentity
\mbox{''}
]$ 

\item $
[EqSysIsPartition
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqSysIsPartition
\mbox{''}
]$

\item $
[
\mathsf{x}
 / 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}/
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 \cap 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}cap
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[\cup 
\mathsf{x}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}cup
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\mbox{''}
]$

\item $
[
\mathsf{x}
 \mathrel{\cup} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}cup
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[P(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}P(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[\{
\mathsf{x}
\}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}
\{
\#1.
\newline 
\mbox {$\backslash $}
\}
\mbox{''}
]$

\item $
[\{
\mathsf{x}
,
\mathsf{y}
\}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}
\{
\#1.
\newline ,
\#2.
\newline 
\mbox {$\backslash $}
\}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
,
\mathsf{y}
\rangle
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline ,
\#2.
\newline 
\mbox {$\backslash $}rangle
\mbox{''}
]$,

\item $
[
\mathsf{x}
 \mathrel{\in} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}in
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{z}
(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#3.
\newline (
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[ReflRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ReflRel(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[SymRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SymRel(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[TransRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TransRel(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$


\item $
[EqRel(
\mathsf {r}
,
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqRel(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$


\item $
[[
\mathsf{x}
 \mathrel{\in} \mathsf {bs}
]_{
\mathsf {r}
}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}[
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}in
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline ]
\_
\{
\#3.
\newline 
\}
\mbox{''}
]$ 

\item $
[Partition(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Partition(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[
\mathsf{x}
\!\mathrel{==}\! 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}!
\mbox {$\backslash $}mathrel
\{
{=}
{=}
\}
\mbox {$\backslash $}!
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
 \mathrel{\subseteq} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}subseteq
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[\dot{\neg}\,(
\mathsf{x}
)n
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}dot
\{
\mbox {$\backslash $}neg
\}
\mbox {$\backslash $},(
\#1.
\newline )n
\mbox{''}
]$

\item $
[
\mathsf{x}
 \mathrel{\notin} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}notin
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 \mathrel{\neq} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}neq
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
\mathrel{\dot{\wedge}} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}dot
\{
\mbox {$\backslash $}wedge
\}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
 \mathrel{\dot{\vee}} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}dot
\{
\mbox {$\backslash $}vee
\}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
\mathrel{\dot{\Leftrightarrow}} 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}dot
\{
\mbox {$\backslash $}Leftrightarrow
\}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[ \{ ph \mathrel{\in} 
\mathsf{x}
\mid 
\mathsf {a}
\}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}
\{
\linebreak [0]\ \hskip0em plus2.0em{}ph
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}mathrel
\{
\mbox {$\backslash $}in
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\mbox {$\backslash $}mid
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline 
\mbox {$\backslash $}
\}
\mbox{''}
]$







\item $
[
\mathsf{x}
\Rightarrow 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(i
\#1.
\newline 
\mbox {$\backslash $}Rightarrow
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline )i
\mbox{''}
]$

\item $
[Nat(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Nat(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[\langle 
\mathsf{x}
{\equiv} 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Me}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Me
\}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
{\equiv}^1 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Me} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}1
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Me
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[\langle 
\mathsf{x}
{\equiv}^* 
\mathsf{y}
| 
\mathsf{z}
{:==} 
\mathsf {u}
\rangle_{Me} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}langle
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\{
\mbox {$\backslash $}equiv
\}
{\char94}
{*}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline |
\linebreak [0]\ \hskip0em plus2.0em{}
\#3.
\newline 
\{
{:}
{=}
{=}
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\#4.
\newline 
\mbox {$\backslash $}rangle
\_
\{Me
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

\item $
[
\exists 
\mathsf{x}
\colon 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\newline 
\mbox {$\backslash $}exists
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\mbox {$\backslash $}colon
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[(x1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(x1)
\mbox{''}
]$ 

\item $
[(x2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(x2)
\mbox{''}
]$

\item $
[(y1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(y1)
\mbox{''}
]$ 

\item $
[(y2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(y2)
\mbox{''}
]$

\item $
[(v1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(v1)
\mbox{''}
]$ 

\item $
[(v2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(v2)
\mbox{''}
]$ 

\item $
[(v3)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(v3)
\mbox{''}
]$

\item $
[(v4)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(v4)
\mbox{''}
]$

\item $
[(v2n)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(v2n)
\mbox{''}
]$ 

\item $
[(n1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(n1)
\mbox{''}
]$ 

\item $
[(n2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(n2)
\mbox{''}
]$ 

\item $
[(n3)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(n3)
\mbox{''}
]$

\item $
[(m1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(m1)
\mbox{''}
]$ 

\item $
[(m2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(m2)
\mbox{''}
]$ 

 \item $
[(\epsilon)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox {$\backslash $}epsilon)
\mbox{''}
]$ 

\item $
[(\epsilon)_{1}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox {$\backslash $}epsilon)
\_
\{1
\}
\mbox{''}
]$ 

\item $
[(\epsilon 2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox {$\backslash $}epsilon
\linebreak [0]\ \hskip0em plus2.0em{}2)
\mbox{''}
]$ 

\item $
[(fx)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fx)
\mbox{''}
]$ 

\item $
[(fy)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fy)
\mbox{''}
]$ 

\item $
[(fz)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fz)
\mbox{''}
]$ 

\item $
[(fu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fu)
\mbox{''}
]$ 

\item $
[(fv)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fv)
\mbox{''}
]$

\item $
[(fw)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fw)
\mbox{''}
]$

\item $
[(fep)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fep)
\mbox{''}
]$

\item $
[(rx)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(rx)
\mbox{''}
]$

\item $
[(ry)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(ry)
\mbox{''}
]$

\item $
[(rz)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(rz)
\mbox{''}
]$

\item $
[(ru)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(ru)
\mbox{''}
]$

\item $
[(sx)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(sx)
\mbox{''}
]$ 

\item $
[(sx1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(sx1)
\mbox{''}
]$ 

\item $
[(sy)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(sy)
\mbox{''}
]$

\item $
[(sy1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(sy1)
\mbox{''}
]$ 

\item $
[(sz)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(sz)
\mbox{''}
]$

\item $
[(sz1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(sz1)
\mbox{''}
]$

\item $
[(su)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(su)
\mbox{''}
]$

\item $
[(su1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(su1)
\mbox{''}
]$

\item $
[(fxs)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fxs)
\mbox{''}
]$

\item $
[(fys)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(fys)
\mbox{''}
]$

\item $
[(crs1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(crs1)
\mbox{''}
]$

\item $
[(f1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(f1)
\mbox{''}
]$ 

\item $
[(f2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(f2)
\mbox{''}
]$ 

\item $
[(f3)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(f3)
\mbox{''}
]$ 

\item $
[(f4)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(f4)
\mbox{''}
]$ 

\item $
[(op1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(op1)
\mbox{''}
]$ 

\item $
[(op2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(op2)
\mbox{''}
]$ 

\item $
[(r1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(r1)
\mbox{''}
]$

\item $
[(s1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(s1)
\mbox{''}
]$

\item $
[(s2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(s2)
\mbox{''}
]$

\item $
[X_{1}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}X
\_
\{1
\}
\mbox{''}
]$ 

\item $
[X_{2}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}X
\_
\{2
\}
\mbox{''}
]$

\item $
[Y_{1}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Y
\_
\{1
\}
\mbox{''}
]$ 

\item $
[Y_{2}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Y
\_
\{2
\}
\mbox{''}
]$

\item $
[V_{1}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}V
\_
\{1
\}
\mbox{''}
]$ 

\item $
[V_{2}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}V
\_
\{2
\}
\mbox{''}
]$

\item $
[V_{3}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}V
\_
\{3
\}
\mbox{''}
]$

\item $
[V_{4}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}V
\_
\{4
\}
\mbox{''}
]$

\item $
[V_{2n}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}V
\_
\{2n
\}
\mbox{''}
]$

\item $
[\epsilon
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}epsilon
\mbox{''}
]$ 

\item $
[M_{1}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}M
\_
\{1
\}
\mbox{''}
]$ 

\item $
[M_{2}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}M
\_
\{2
\}
\mbox{''}
]$

\item $
[N_{1} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}N
\_
\{1
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$ 

\item $
[N_{2} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}N
\_
\{2
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$ 

\item $
[N_{3} 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}N
\_
\{3
\}
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$ 

\item $
[\epsilon 1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}epsilon
\linebreak [0]\ \hskip0em plus2.0em{}1
\mbox{''}
]$ 

\item $
[\epsilon 2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox {$\backslash $}epsilon
\linebreak [0]\ \hskip0em plus2.0em{}2
\mbox{''}
]$ 

\item $
[FX
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FX
\mbox{''}
]$ 

\item $
[FY
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FY
\mbox{''}
]$ 

\item $
[FZ
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FZ
\mbox{''}
]$

\item $
[FU
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FU
\mbox{''}
]$

\item $
[FV
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FV
\mbox{''}
]$

\item $
[FW
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FW
\mbox{''}
]$

\item $
[FEP
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FEP
\mbox{''}
]$

\item $
[RX
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}RX
\mbox{''}
]$

\item $
[RY
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}RY
\mbox{''}
]$ 

\item $
[RZ
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}RZ
\mbox{''}
]$

\item $
[RU
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}RU
\mbox{''}
]$

\item $
[(SX)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(SX)
\mbox{''}
]$ 

\item $
[(SX1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(SX1)
\mbox{''}
]$ 

\item $
[(SY)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(SY)
\mbox{''}
]$

\item $
[(SY1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(SY1)
\mbox{''}
]$ 

\item $
[(SZ)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(SZ)
\mbox{''}
]$

\item $
[(SZ1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(SZ1)
\mbox{''}
]$

\item $
[(SU)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(SU)
\mbox{''}
]$

\item $
[(SU1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(SU1)
\mbox{''}
]$

\item $
[FXS
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FXS
\mbox{''}
]$

\item $
[FYS
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FYS
\mbox{''}
]$

\item $
[(F1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(F1)
\mbox{''}
]$ 

\item $
[(F2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(F2)
\mbox{''}
]$ 

\item $
[(F3)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(F3)
\mbox{''}
]$ 

\item $
[(F4)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(F4)
\mbox{''}
]$ 

\item $
[(OP1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(OP1)
\mbox{''}
]$ 

\item $
[(OP2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(OP2)
\mbox{''}
]$ 

\item $
[(R1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(R1)
\mbox{''}
]$ 

\item $
[(S1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(S1)
\mbox{''}
]$

\item $
[(S2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(S2)
\mbox{''}
]$

\item $
[(EPob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(EPob)
\mbox{''}
]$ 

\item $
[(CRS1ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(CRS1ob)
\mbox{''}
]$ 

\item $
[(F1ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(F1ob)
\mbox{''}
]$ 

\item $
[(F2ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(F2ob)
\mbox{''}
]$ 

\item $
[(F3ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(F3ob)
\mbox{''}
]$ 

\item $
[(F4ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(F4ob)
\mbox{''}
]$ 

\item $
[(N1ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(N1ob)
\mbox{''}
]$ 

\item $
[(N2ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(N2ob)
\mbox{''}
]$

\item $
[(OP1ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(OP1ob)
\mbox{''}
]$ 

\item $
[(OP2ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(OP2ob)
\mbox{''}
]$ 

\item $
[(R1ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(R1ob)
\mbox{''}
]$ 

\item $
[(S1ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(S1ob)
\mbox{''}
]$

\item $
[(S2ob)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(S2ob)
\mbox{''}
]$






\item $
[Ex3
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Ex3
\mbox{''}
]$ 

\item $
[NAT
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NAT
\mbox{''}
]$ 

\item $
[RATIONAL_SERIES
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}RATIONAL
\_SERIES
\mbox{''}
]$

\item $
[SERIES
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SERIES
\mbox{''}
]$

\item $
[SetOfReals
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SetOfReals
\mbox{''}
]$ 

\item $
[SetOfFxs
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SetOfFxs
\mbox{''}
]$

\item $
[N
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}N
\mbox{''}
]$ 

\item $
[Q
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Q
\mbox{''}
]$ 

\item $
[X
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}X
\mbox{''}
]$ 

\item $
[xs
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}xs
\mbox{''}
]$ 

\item $
[xaF
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}xaF
\mbox{''}
]$

\item $
[ysF
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ysF
\mbox{''}
]$

\item $
[us
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}us
\mbox{''}
]$

\item $
[usFoelge
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}usFoelge
\mbox{''}
]$

\item $
[0
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0
\mbox{''}
]$ 

\item $
[1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}1
\mbox{''}
]$

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

\item $
[2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}2
\mbox{''}
]$

\item $
[3
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}3
\mbox{''}
]$

\item $
[1/2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}1/2
\mbox{''}
]$

\item $
[1/3
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}1/3
\mbox{''}
]$

\item $
[2/3
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}2/3
\mbox{''}
]$

\item $
[0f
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0f
\mbox{''}
]$ 

\item $
[00
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}00
\mbox{''}
]$ 

\item $
[(--01)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox{-}
\mbox{-}01)
\mbox{''}
]$ 

\item $
[02
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}02
\mbox{''}
]$ 

\item $
[01//02
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}01//02
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 = 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 \neq 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox {$\backslash $}neq
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
 < 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 <= 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
<_{f}
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
{<}
\_
\{f
\}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
\leq_{f}
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox {$\backslash $}leq
\_
\{f
\}
\#2.
\mbox{''}
]$

\item $
[SF(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SF(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 == 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{=}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
 !!== 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}!!
{=}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
 << 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
{<}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
 <<== 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}
{<}
{<}
{=}
{=}
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$ 

\item $
[
\mathsf{x}
[
\mathsf{y}
]
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline [
\#2.
\newline ]
\mbox{''}
]$ 

\item $
[(-u
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox{-}u
\#1.
\newline )
\mbox{''}
]$ 

\item $
[-_{f}
\mathsf{x}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox{-}
\_
\{f
\}
\#1.
\mbox{''}
]$

\item $
[(--
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox{-}
\mbox{-}
\#1.
\newline )
\mbox{''}
]$

\item $
[1f/
\mathsf{x}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}1f/
\#1.
\mbox{''}
]$

\item $
[01//temp
\mathsf{x}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}01//temp
\#1.
\mbox{''}
]$ 

\item $
[(
\mathsf{x}
+
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\#1.
\newline 
{+}
\#2.
\newline )
\mbox{''}
]$ 

\item $
[(
\mathsf{x}
-
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\#1.
\newline 
\mbox{-}
\#2.
\newline )
\mbox{''}
]$ 

\item $
[(fx)
+_{f}(fy)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
{+}
\_
\{f
\}
\#2.
\mbox{''}
]$

\item $
[(fx)
-_{f}(fy)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
\mbox{-}
\_
\{f
\}
\#2.
\mbox{''}
]$

\item $
[(fx)
*_{f}(fy)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
{*}
\_
\{f
\}
\#2.
\mbox{''}
]$

\item $
[
\mathsf{x}
++
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
{+}
{+}
\#2.
\mbox{''}
]$

\item $
[R((fx)
) -- R((fy)
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}R(
\#1.
\newline )
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}
\mbox{-}
\linebreak [0]\ \hskip0em plus2.0em{}R(
\#2.
\newline )
\mbox{''}
]$ 

\item $
[(
\mathsf{x}
*
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\#1.
\newline 
{*}
\#2.
\newline )
\mbox{''}
]$

\item $
[
\mathsf{x}
**
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\#1.
\newline 
{*}
{*}
\#2.
\mbox{''}
]$

\item $
[ 
\mathsf{x}
 (exp) 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\linebreak [0]\ \hskip0em plus2.0em{}
\#1.
\newline 
\linebreak [0]\ \hskip0em plus2.0em{}(exp)
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\mbox{''}
]$

\item $
[leqReflexivity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}leqReflexivity
\mbox{''}
]$ 

\item $
[rec
\mathsf{x}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}rec
\#1.
\mbox{''}
]$

\item $
[|
\mathsf{x}
|
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}|
\#1.
\newline |
\mbox{''}
]$ 

\item $
[StateExpand(
\mathsf {t}
,
\mathsf {s}
,
\mathsf {c}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}StateExpand(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$ 

\item $
[extractSeries(
\mathsf {t}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}extractSeries(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[|f
\mathsf{x}
|
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}|f
\#1.
\newline |
\mbox{''}
]$

\item $
[|r
\mathsf{x}
|
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}|r
\#1.
\newline |
\mbox{''}
]$ 

\item $
[SetOfSeries(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SetOfSeries(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[ExpandList(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpandList(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$ 

\item $
[**Macro(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
{*}
{*}Macro(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[++Macro(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
{+}
{+}Macro(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[--Macro(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox{-}
\mbox{-}Macro(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[<<Macro(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
{<}
{<}Macro(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[||Macro(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}||Macro(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[01//Macro(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}01//Macro(
\#1.
\newline )
\mbox{''}
]$ 



\item $
[Max(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Max(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[Max(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Max(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$

\item $
[Limit(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Limit(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[Union(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Union(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[if(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}if(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$ 

\item $
[IsOrderedPair(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IsOrderedPair(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$ 

\item $
[IsRelation(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IsRelation(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$ 

\item $
[isFunction(
\mathsf{x}
,
\mathsf{y}
,
\mathsf{z}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}isFunction(
\#1.
\newline ,
\#2.
\newline ,
\#3.
\newline )
\mbox{''}
]$ 

\item $
[TypeNat(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TypeNat(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[TypeNat0(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TypeNat0(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[TypeRational(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TypeRational(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[TypeRational0(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TypeRational0(
\#1.
\newline )
\mbox{''}
]$

\item $
[TypeSeries(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TypeSeries(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[Typeseries0(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Typeseries0(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[UB(
\mathsf{x}
, 
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UB(
\#1.
\newline ,
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline )
\mbox{''}
]$ 

\item $
[LUB(
\mathsf{x}
, 
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LUB(
\#1.
\newline ,
\linebreak [0]\ \hskip0em plus2.0em{}
\#2.
\newline )
\mbox{''}
]$ 

\item $
[BS(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BS(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[UStelescope(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UStelescope(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[R(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}R(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[--R(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox{-}
\mbox{-}R(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[IsSeries(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IsSeries(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[IsNatural(
\mathsf{x}
\mathsf{y}
,\ast
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IsNatural(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[OrderedPair(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}OrderedPair(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$


\item $
[leqAntisymmetryAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}leqAntisymmetryAxiom
\mbox{''}
]$

\item $
[leqTransitivityAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}leqTransitivityAxiom
\mbox{''}
]$

\item $
[leqTotality
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}leqTotality
\mbox{''}
]$

\item $
[leqAdditionAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}leqAdditionAxiom
\mbox{''}
]$

\item $
[leqMultiplicationAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}leqMultiplicationAxiom
\mbox{''}
]$

\item $
[plusAssociativity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}plusAssociativity
\mbox{''}
]$ 

\item $
[plusCommutativity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}plusCommutativity
\mbox{''}
]$

\item $
[Negative
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Negative
\mbox{''}
]$

\item $
[plus0
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}plus0
\mbox{''}
]$

\item $
[timesAssociativity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}timesAssociativity
\mbox{''}
]$ 

\item $
[timesCommutativity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}timesCommutativity
\mbox{''}
]$

\item $
[ReciprocalAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ReciprocalAxiom
\mbox{''}
]$

\item $
[times1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}times1
\mbox{''}
]$

\item $
[plusAssociativity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}plusAssociativity
\mbox{''}
]$ 

\item $
[plusCommutativity
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}plusCommutativity
\mbox{''}
]$

\item $
[Negative
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Negative
\mbox{''}
]$

\item $
[Distribution
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Distribution
\mbox{''}
]$

\item $
[0not1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0not1
\mbox{''}
]$

\item $
[A4(Axiom)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}A4(Axiom)
\mbox{''}
]$ 

\item $
[InductionAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}InductionAxiom
\mbox{''}
]$

\item $
[EqualityAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqualityAxiom
\mbox{''}
]$

\item $
[EqLeqAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqLeqAxiom
\mbox{''}
]$

\item $
[EqAdditionAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqAdditionAxiom
\mbox{''}
]$

\item $
[EqMultiplicationAxiom
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqMultiplicationAxiom
\mbox{''}
]$
                                         
\item $
[SENC1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SENC1
\mbox{''}
]$ 

\item $
[SENC2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SENC2
\mbox{''}
]$ 

\item $
[Cauchy
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Cauchy
\mbox{''}
]$ 

\item $
[PlusF
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PlusF
\mbox{''}
]$ 

\item $
[ReciprocalF
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ReciprocalF
\mbox{''}
]$

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

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


\item $
[FromInR
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromInR
\mbox{''}
]$ 

\item $
[ReciprocalR(Axiom)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ReciprocalR(Axiom)
\mbox{''}
]$

\item $
[US0
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}US0
\mbox{''}
]$ 

\item $
[NextXS(UpperBound)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NextXS(UpperBound)
\mbox{''}
]$ 

\item $
[NextXS(NoUpperBound)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NextXS(NoUpperBound)
\mbox{''}
]$ 

\item $
[NextUS(UpperBound)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NextUS(UpperBound)
\mbox{''}
]$ 

\item $
[NextUS(NoUpperBound)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NextUS(NoUpperBound)
\mbox{''}
]$

\item $
[ExpZero
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpZero
\mbox{''}
]$ 

\item $
[ExpPositive
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpPositive
\mbox{''}
]$

\item $
[ExpZero(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpZero(R)
\mbox{''}
]$ 

\item $
[ExpPositive(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpPositive(R)
\mbox{''}
]$

\item $
[LessMinus1(N)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessMinus1(N)
\mbox{''}
]$

\item $
[Nonnegative(N)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Nonnegative(N)
\mbox{''}
]$ 

\item $
[BSzero
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BSzero
\mbox{''}
]$ 

\item $
[BSpositive
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BSpositive
\mbox{''}
]$ 

\item $
[UStelescope(Zero)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UStelescope(Zero)
\mbox{''}
]$ 

\item $
[UStelescope(Positive)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UStelescope(Positive)
\mbox{''}
]$

\item $
[EqAddition(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqAddition(R)
\mbox{''}
]$ 

\item $
[FromLimit
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromLimit
\mbox{''}
]$ 

\item $
[ToUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToUpperBound
\mbox{''}
]$ 

\item $
[FromUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromUpperBound
\mbox{''}
]$ 

\item $
[USisUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USisUpperBound
\mbox{''}
]$ 

\item $
[0not1(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0not1(R)
\mbox{''}
]$ 

\item $
[ExpUnbounded(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpUnbounded(R)
\mbox{''}
]$ 

\item $
[FromLeq(Advanced)(N)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromLeq(Advanced)(N)
\mbox{''}
]$ 

\item $
[FromLeastUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromLeastUpperBound
\mbox{''}
]$ 

\item $
[ToLeastUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToLeastUpperBound
\mbox{''}
]$ 

\item $
[XSisNotUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}XSisNotUpperBound
\mbox{''}
]$ 

\item $
[ysFGreater
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ysFGreater
\mbox{''}
]$ 

\item $
[ysFLess
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ysFLess
\mbox{''}
]$ 

\item $
[SmallInverse
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SmallInverse
\mbox{''}
]$ 

\item $
[MemberOfSeries(Imply)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MemberOfSeries(Imply)
\mbox{''}
]$ 

\item $
[NatType
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NatType
\mbox{''}
]$ 

\item $
[RationalType
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}RationalType
\mbox{''}
]$

\item $
[SeriesType
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SeriesType
\mbox{''}
]$ 


\item $
[JoinConjuncts(2conditions)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}JoinConjuncts(2conditions)
\mbox{''}
]$ 

\item $
[TND
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TND
\mbox{''}
]$ 

\item $
[FromNegatedImply
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromNegatedImply
\mbox{''}
]$

\item $
[ToNegatedImply
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToNegatedImply
\mbox{''}
]$ 

\item $
[FromNegated(2*Imply)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromNegated(2
{*}Imply)
\mbox{''}
]$ 

\item $
[FromNegatedAnd
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromNegatedAnd
\mbox{''}
]$

\item $
[FromNegatedOr
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromNegatedOr
\mbox{''}
]$ 

\item $
[ToNegatedOr
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToNegatedOr
\mbox{''}
]$

\item $
[FromNegations
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromNegations
\mbox{''}
]$ 

\item $
[From3Disjuncts
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}From3Disjuncts
\mbox{''}
]$ 

\item $
[NegateDisjunct1
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegateDisjunct1
\mbox{''}
]$ 

\item $
[NegateDisjunct2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegateDisjunct2
\mbox{''}
]$ 

\item $
[ExpandDisjuncts
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpandDisjuncts
\mbox{''}
]$ 

\item $
[From2*2Disjuncts
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}From2
{*}2Disjuncts
\mbox{''}
]$ 







\item $
[PlusR(Sym)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PlusR(Sym)
\mbox{''}
]$ 

\item $
[LessLeq(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessLeq(R)
\mbox{''}
]$ 


\item $
[LeqAntisymmetry(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LeqAntisymmetry(R)
\mbox{''}
]$ 

\item $
[LeqTransitivity(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LeqTransitivity(R)
\mbox{''}
]$

\item $
[Plus0(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Plus0(R)
\mbox{''}
]$


\item $
[lessAddition(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}lessAddition(R)
\mbox{''}
]$ 

 


\item $
[leqAddition(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}leqAddition(R)
\mbox{''}
]$ 

\item $
[PlusAssociativity(R)XX
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PlusAssociativity(R)XX
\mbox{''}
]$

\item $
[PlusAssociativity(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PlusAssociativity(R)
\mbox{''}
]$

\item $
[Negative(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Negative(R)
\mbox{''}
]$ 

\item $
[PlusCommutativity(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PlusCommutativity(R)
\mbox{''}
]$



\item $
[Times1(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Times1(R)
\mbox{''}
]$

\item $
[TimesAssociativity(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TimesAssociativity(R)
\mbox{''}
]$

 


\item $
[TimesCommutativity(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TimesCommutativity(R)
\mbox{''}
]$

\item $
[Distribution(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Distribution(R)
\mbox{''}
]$ 

 \item $
[
\exists 
\mathsf{x}
\colon 
\mathsf{y}
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(AARRGGHH!
\mbox{-}exist
\mbox{-}bug!)
\mbox{''}
]$   

\item $
[constantRationalSeries(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}constantRationalSeries(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[Power(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Power(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[cartProd(
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}cartProd(
\#1.
\newline )
\mbox{''}
]$ 

\item $
[binaryUnion(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}binaryUnion(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[SetOfRationalSeries
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SetOfRationalSeries
\mbox{''}
]$ 

\item $
[MemberOfSeries
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MemberOfSeries
\mbox{''}
]$ 

\item $
[IsSubset(
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IsSubset(
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[memberOfSeries(Type)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}memberOfSeries(Type)
\mbox{''}
]$ 

\item $
[UniqueMember
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UniqueMember
\mbox{''}
]$ 

\item $
[UniqueMember(Type)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UniqueMember(Type)
\mbox{''}
]$

\item $
[SameSeries
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameSeries
\mbox{''}
]$ 

\item $
[A4
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}A4
\mbox{''}
]$ 

\item $
[(s
\mathsf{x}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(s
\#1.
\newline )
\mbox{''}
]$ 

\item $
[(p
\mathsf{x}
,
\mathsf{y}
)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(p
\#1.
\newline ,
\#2.
\newline )
\mbox{''}
]$ 

\item $
[SameMember
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameMember
\mbox{''}
]$ 

\item $
[Qclosed(Addition)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Qclosed(Addition)
\mbox{''}
]$ 

\item $
[Qclosed(Multiplication)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Qclosed(Multiplication)
\mbox{''}
]$

\item $
[FromCartProd(1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromCartProd(1)
\mbox{''}
]$ 

\item $
[FromCartProd(1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromCartProd(1)
\mbox{''}
]$


\item $
[Max
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Max
\mbox{''}
]$ 

\item $
[Numerical
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Numerical
\mbox{''}
]$ 

\item $
[NumericalF
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NumericalF
\mbox{''}
]$ 

\item $
[Separation2formula(1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Separation2formula(1)
\mbox{''}
]$ 

\item $
[Separation2formula(2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Separation2formula(2)
\mbox{''}
]$

\item $
[QisClosed(Reciprocal)(Imply)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}QisClosed(Reciprocal)(Imply)
\mbox{''}
]$ 

\item $
[QisClosed(Reciprocal)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}QisClosed(Reciprocal)
\mbox{''}
]$ 

\item $
[QisClosed(Negative)(Imply)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}QisClosed(Negative)(Imply)
\mbox{''}
]$ 

\item $
[QisClosed(Negative)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}QisClosed(Negative)
\mbox{''}
]$ 

\item $
[(Adgic)SameR
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(Adgic)SameR
\mbox{''}
]$ 

\end{list}


\end{document}

