\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Ewords\_2EWORD\_\_NEG (TMQS5uHYRyYAsVindEHDzdLSvLE5RqWK44D)}
\maketitle
Let $ ty\_2Efcp\_2Efinite\_\_image : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Efcp\_2Efinite\_\_image}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfeggjgogjgehfgpfpfjgngbghgfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
Let $ ty\_2Ebool\_2Eitself : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ebool\_2Eitself}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfecgpgpgmgpfcdfejgehdhfgmggg}
\end{equation}
Let $ c\_2Ebool\_2Ethe\_\_value : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebool\_2Ethe\_\_value~A\_27a \in (\\
ty\_2Ebool\_2Eitself~A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfeehigfgpfpfghbgmgfhfg}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Efcp\_2Edimindex : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Efcp\_2Edimindex~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegjgngjgogegfgih}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddgepfcdbd}
We define $\mname{c\_2Ebool\_2E\_3F\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{V0P}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V0P}}~{V1x}})}})}}~{({{{ap}~{V0P}}~{V2y}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfeggjgogjgehfgpfpfjgogegfgih}
We define $\mname{c\_2Efcp\_2Efinite\_\_index}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V0f \in {({{A\_27a}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2n}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{V0f}}~{V2n}})}})}}~{V1x}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ ty\_2Efcp\_2Ecart : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Efcp\_2Ecart~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfedgbgcheh}
\end{equation}
Let $ c\_2Efcp\_2Edest\_\_cart : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Efcp\_2Edest\_\_cart~\\
A\_27a~A\_27b \in ((A\_27a^{{({{ty\_2Efcp\_2Efinite\_\_image}~{A\_27b}})}})^{{({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegfgdhehpfpfdgbgcheh}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfeggdgahpfpfjgogegfgih}
We define $\mname{c\_2Efcp\_2Efcp\_\_index}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{c\_2Efcp\_2Edest\_\_cart}~{A\_27a}}~{A\_27b}})}}~{V0x}})}}~{({{{ap}~{({{\mname{c\_2Efcp\_2Efinite\_\_index}}~{({{ty\_2Efcp\_2Efinite\_\_image}~{A\_27b}})}})}}~{V1i}})}})}}}}}$.
\end{definition}
Let $ c\_2Earithmetic\_2EEVEN : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EEVEN} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefegffeoe}
\end{equation}
Let $ c\_2Earithmetic\_2EODD : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EODD} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepeeeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfe}
We define $\mname{c\_2Earithmetic\_2E\_3E}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V1n}})}}~{V0m}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3E\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3E}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\end{definition}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfeafcffe}
We define $\mname{c\_2Eprim\_\_rec\_2EPRE}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{\mname{c\_2Enum\_2E0}}})}})}}~{\mname{c\_2Enum\_2E0}}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V1n}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2A : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2A \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdbe}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfcdfejgjgdfffde}
We define $\mname{c\_2Enumeral\_2EiiSUC}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V0n}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfcdfejgkf}
We define $\mname{c\_2Enumeral\_2EiZ}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfdddepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3C\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2B : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2B \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdce}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefcd}
We define $\mname{c\_2Earithmetic\_2EBIT2}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2EEXP : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EEXP \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefeifaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfedfcejeef}
We define $\mname{c\_2Ebit\_2ESBIT}$ to be ${{\uniddcgcg} V0b \in {2} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{V0b}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V1n}})}})}}~{\mname{c\_2Enum\_2E0}}})}}}$.
\end{definition}
Let $ c\_2Esum\_\_num\_2ESUM : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Esum\_\_num\_2ESUM \in ((ty\_2Enum\_2Enum^{{({{ty\_2Enum\_2Enum}^{{ty\_2Enum\_2Enum}}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfpfogfhngpfcdfedfffne}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhcdog}
We define $\mname{c\_2Ewords\_2Ew2n}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{c\_2Esum\_\_num\_2ESUM}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2ESBIT}}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0w}})}}~{V1i}})}})}}~{V1i}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefbd}
We define $\mname{c\_2Earithmetic\_2EBIT1}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2EDIV : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EDIV \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfeeejegf}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfeeejegfpfpfcdfeifaf}
We define $\mname{c\_2Ebit\_2EDIV\_\_2EXP}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EDIV}}~{V1n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V0x}})}})}}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2D : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2D \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdee}
\end{equation}
Let $ c\_2Earithmetic\_2EMOD : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EMOD \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfenepeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfenepeeepfpfcdfeifaf}
We define $\mname{c\_2Ebit\_2EMOD\_\_2EXP}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EMOD}}~{V1n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfecejeefdf}
We define $\mname{c\_2Ebit\_2EBITS}$ to be ${{\uniddcgcg} V0h \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1l \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EMOD\_\_2EXP}}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V0h}})}})}}~{V1l}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EDIV\_\_2EXP}}}~{V1l}})}}~{V2n}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfecejeef}
We define $\mname{c\_2Ebit\_2EBIT}$ to be ${{\uniddcgcg} V0b \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EBITS}}}~{V0b}})}}~{V0b}})}}~{V1n}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfegedeaf}
We define $\mname{c\_2Efcp\_2EFCP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0g \in {({{A\_27a}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})}})}}~{({{\uniddcgcg} V1f \in {({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2i}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27b}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27b}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{A\_27a}}~{A\_27b}})}}~{V1f}})}}~{V2i}})}})}}~{({{{ap}~{V0g}}~{V2i}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfeogcdhh}
We define $\mname{c\_2Ewords\_2En2w}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebit\_2EBIT}}}~{V1i}})}}~{V0n}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfbgegeg}
We define $\mname{c\_2Ewords\_2Eword\_\_add}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0v \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {{\uniddcgcg} V1w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27a}})}}~{V0v}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27a}})}}~{V1w}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfbddgpgngah}
We define $\mname{c\_2Ewords\_2Eword\_\_1comp}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0w}})}}~{V1i}})}})}})}})}}}$.
\end{definition}
Let $ c\_2Ewords\_2Edimword : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewords\_2Edimword~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhpgchegdhpfcdfeegjgnghhpgcheg}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfcddgpgngah}
We define $\mname{c\_2Ewords\_2Eword\_\_2comp}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ewords\_2En2w}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2D}}~{({{{ap}~{({{c\_2Ewords\_2Edimword}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Ew2n}}~{A\_27a}})}}~{V0w}})}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~(ap~c\_2Earithmetic\_2E\_2B~\mname{c\_2Enum\_2E0})~V0m)=V0m){\unicdcdcdhd}(((ap~(\\
ap~c\_2Earithmetic\_2E\_2B~V0m)~\mname{c\_2Enum\_2E0})=V0m){\unicdcdcdhd}(((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0m))~V1n)=(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0m)~V1n))){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n))=(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n)=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1n)~V0m))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfdfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V1n)~V2p))=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))~V2p)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V1n)){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0m))~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
\mname{c\_2Enum\_2E0})~V0n)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpepfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~(ap~c\_2Earithmetic\_2E\_2A~\mname{c\_2Enum\_2E0})~V0m)=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~V0m)=V0m){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))=V0m){\unicdcdcdhd}(\\
((ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Enum\_2ESUC}~V0m))~V1n)=(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~(ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~V1n))~\\
V1n)){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~(ap~\mname{c\_2Enum\_2ESUC}~V1n))=\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V0m)~V1n))))))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeneffmeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  (((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~V1n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1n)~V2p))){\unicdbdjdcd}(p~(\\
ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(V0m=V1n){\unicdbdjded}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V1n)){\unicdcdcdhd}(p~(\\
ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1n)~V0m))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfefebfpfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0m)~V2p))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1n)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfnepeoepepfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(\preoh (V0m=V1n)){\unicdbdjded}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Enum\_2ESUC}~\\
V0m))~V1n)){\unicdcdcdid}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n))~V0m))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepeefpfpfoeffnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Enum\_2ESUC}~V0n)=(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))~V0n)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffdepfpfpeoefepfpfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~(ap~\\
(ap~c\_2Earithmetic\_2E\_2D~V0m)~V1n))~V2p)=(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V0m)~(ap~(ap~c\_2Earithmetic\_2E\_2B~V1n)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpfcfjeheieefpfpfdfffce}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1k \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1k)~V0n)){\unicdbdjdcd}((ap~(ap~c\_2Earithmetic\_2EMOD~\\
V1k)~V0n)=V1k))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfnepeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0P \in (2^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1a \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V2b \in ty\_2Enum\_2Enum .  ((p~(ap~V0P~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V1a)~V2b))){\unicdbdjded}({\unicdcdadad} V3d \in ty\_2Enum\_2Enum .  (((V2b=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1a)~V3d)){\unicdbdjdcd}(p~(ap~V0P~\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}((V1a=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V2b)~V3d)){\unicdbdjdcd}(p~(ap~V0P~V3d)))))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpffemejenepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdcdcdhd}\\
((p~V1t2){\unicdcdcdhd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdcdcdhd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoekepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  (((p~V0A){\unicdcdcdid}(\\
(p~V1B){\unicdcdcdid}(p~V2C))){\unicdbdjded}(((p~V0A){\unicdcdcdid}(p~V1B)){\unicdcdcdid}(p~V2C))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeejedfkepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdhd}(p~V1B))){\unicdbdjded}((\preoh (\\
p~V0A)){\unicdcdcdid}(\preoh (p~V1B)))){\unicdcdcdhd}((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjded}((\preoh (p~V0A)){\unicdcdcdhd}(\preoh (p~V1B)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeefepfpfnepecfhebeoepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((p~V0A){\unicdbdjdcd}(p~V1B)){\unicdbdjded}((\preoh (p~V0A)){\unicdcdcdid}\\
(p~V1B)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfeejedfkepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({{({{p}~{V0t}})}{\unicdbdjdcd}{False}})}{\unicdbdjded}{({{({{p}~{V0t}})}{\unicdbdjded}{False}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfgepfpffebfpfpfge}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1x\_27 \in 2 .  ({\unicdcdadad} V2y \in 2 .  ({\unicdcdadad} V3y\_27 \in \\
2 .  ((((p~V0x){\unicdbdjded}(p~V1x\_27)){\unicdcdcdhd}((p~V1x\_27){\unicdbdjdcd}((p~V2y){\unicdbdjded}(p~V3y\_27)))){\unicdbdjdcd}\\
(((p~V0x){\unicdbdjdcd}(p~V2y)){\unicdbdjded}((p~V1x\_27){\unicdbdjdcd}(p~V3y\_27))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in (ty\_2Efcp\_2Ecart~A\_27a~A\_27b) .  ({\unicdcdadad} V1y \in (ty\_2Efcp\_2Ecart~\\
A\_27a~A\_27b) .  ((V0x=V1y){\unicdbdjded}({\unicdcdadad} V2i \in ty\_2Enum\_2Enum .  ((p~(ap~\\
(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V2i)~(ap~(c\_2Efcp\_2Edimindex~A\_27b)~(\\
c\_2Ebool\_2Ethe\_\_value~A\_27b)))){\unicdbdjdcd}((ap~(ap~(\mname{c\_2Efcp\_2Efcp\_\_index}~\\
A\_27a~A\_27b)~V0x)~V2i)=(ap~(ap~(\mname{c\_2Efcp\_2Efcp\_\_index}~A\_27a~A\_27b)~\\
V1y)~V2i)))))))\end{array}
\label{hyp:ehigngpfcdfeggdgahpfcdfedebecfefpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
\mname{c\_2Enum\_2E0})~V0n)=V0n)){\unicdcdcdhd}(({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V1n)~\mname{c\_2Enum\_2E0})=V1n)){\unicdcdcdhd}(({\unicdcdadad} V2n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V3m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V2n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V3m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Enumeral\_2EiZ}~(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V2n)~V3m)))))){\unicdcdcdhd}(({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2E\_2A~\mname{c\_2Enum\_2E0})~V4n)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}\\
(({\unicdcdadad} V5n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V5n)~\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V6n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V7m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~(\\
ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V6n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V7m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V6n)~V7m))))){\unicdcdcdhd}(({\unicdcdadad} V8n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
\mname{c\_2Enum\_2E0})~V8n)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V9n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2E\_2D~V9n)~\mname{c\_2Enum\_2E0})=V9n)){\unicdcdcdhd}(({\unicdcdadad} V10n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V11m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V10n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V11m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V10n)~V11m))))){\unicdcdcdhd}(({\unicdcdadad} V12n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~\\
\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V12n)))=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V13n \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~c\_2Earithmetic\_2EEXP~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V13n)))=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V14n \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~V14n)~\mname{c\_2Enum\_2E0})=\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}\\
(({\unicdcdadad} V15n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V16m \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2EEXP~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V15n))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V16m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~(ap~c\_2Earithmetic\_2EEXP~V15n)~V16m))))){\unicdcdcdhd}(((ap~\mname{c\_2Enum\_2ESUC}~\\
\mname{c\_2Enum\_2E0})=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V17n \in ty\_2Enum\_2Enum .  (\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V17n))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V17n)))){\unicdcdcdhd}(((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\mname{c\_2Enum\_2E0})=\\
\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(({\unicdcdadad} V18n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V18n))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~V18n)))){\unicdcdcdhd}(({\unicdcdadad} V19n \in ty\_2Enum\_2Enum .  \\
(((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V19n)=\mname{c\_2Enum\_2E0}){\unicdbdjded}(V19n=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}\\
(({\unicdcdadad} V20n \in ty\_2Enum\_2Enum .  ((\mname{c\_2Enum\_2E0}=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V20n)){\unicdbdjded}(V20n=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V21n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V22m \in ty\_2Enum\_2Enum .  (((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V21n)=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V22m)){\unicdbdjded}(V21n=V22m)))){\unicdcdcdhd}\\
(({\unicdcdadad} V23n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V23n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}False)){\unicdcdcdhd}(({\unicdcdadad} V24n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V24n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Earithmetic\_2EZERO})~\\
V24n)))){\unicdcdcdhd}(({\unicdcdadad} V25n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V26m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V25n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V26m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V25n)~V26m))))){\unicdcdcdhd}(({\unicdcdadad} V27n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~\\
\mname{c\_2Enum\_2E0})~V27n)){\unicdbdjded}False)){\unicdcdcdhd}(({\unicdcdadad} V28n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V28n))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Earithmetic\_2EZERO})~\\
V28n)))){\unicdcdcdhd}(({\unicdcdadad} V29n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V30m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V29n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V30m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V30m)~V29n))))){\unicdcdcdhd}(({\unicdcdadad} V31n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
\mname{c\_2Enum\_2E0})~V31n)){\unicdbdjded}True)){\unicdcdcdhd}(({\unicdcdadad} V32n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V32n))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V32n)~\\
\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}(({\unicdcdadad} V33n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V34m \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V33n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V34m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V33n)~V34m))))){\unicdcdcdhd}(\\
({\unicdcdadad} V35n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\\
V35n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}True)){\unicdcdcdhd}(({\unicdcdadad} V36n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\mname{c\_2Enum\_2E0})~V36n)){\unicdbdjded}(V36n=\\
\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(({\unicdcdadad} V37n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V38m \in \\
ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V37n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V38m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V38m)~V37n))))){\unicdcdcdhd}(({\unicdcdadad} V39n \in ty\_2Enum\_2Enum .  ((p~(ap~c\_2Earithmetic\_2EODD~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V39n))){\unicdbdjded}(p~(ap~c\_2Earithmetic\_2EODD~\\
V39n)))){\unicdcdcdhd}(({\unicdcdadad} V40n \in ty\_2Enum\_2Enum .  ((p~(ap~c\_2Earithmetic\_2EEVEN~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V40n))){\unicdbdjded}(p~(ap~c\_2Earithmetic\_2EEVEN~\\
V40n)))){\unicdcdcdhd}((\preoh (p~(ap~c\_2Earithmetic\_2EODD~\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(p~(\\
ap~c\_2Earithmetic\_2EEVEN~\mname{c\_2Enum\_2E0})))))))))))))))))))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfegjgdhehchjgcg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  (\\
((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\mname{c\_2Earithmetic\_2EZERO})~\\
V0n))=V0n){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~\mname{c\_2Earithmetic\_2EZERO}))=V0n){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiZ}~(\\
ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V0n))~(\\
ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~(\\
ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}\\
(((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\mname{c\_2Earithmetic\_2EZERO})~\\
V0n))=(ap~\mname{c\_2Enum\_2ESUC}~V0n)){\unicdcdcdhd}(((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~\mname{c\_2Earithmetic\_2EZERO}))=(ap~\mname{c\_2Enum\_2ESUC}~V0n)){\unicdcdcdhd}(((ap~\\
\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
(ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}\\
(((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\mname{c\_2Earithmetic\_2EZERO})~\\
V0n))=(ap~\mname{c\_2Enumeral\_2EiiSUC}~V0n)){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiiSUC}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~\mname{c\_2Earithmetic\_2EZERO}))=(\\
ap~\mname{c\_2Enumeral\_2EiiSUC}~V0n)){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(\\
ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~V1m)))){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~(ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~V1m)))){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~(ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~V1m)))){\unicdcdcdhd}((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~(ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~V1m)))))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfbgegeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  (\\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\mname{c\_2Earithmetic\_2EZERO})~V0n)){\unicdbdjded}\\
True){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~\mname{c\_2Earithmetic\_2EZERO})){\unicdbdjded}False){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V0n))~\mname{c\_2Earithmetic\_2EZERO})){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m))){\unicdbdjded}(\preoh (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1m)~V0n)))){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m)))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfmgehfg}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0t}})}}})}}})}{\unicdbdjded}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeoepeefpfpfoepeef}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{p}~{V0A}})}{\unicdbdjdcd}{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfpfpfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
(((p~V0A){\unicdbdjdcd}False){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((\preoh (p~V0A)){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
((p~V0A){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemedd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{({{({{({{p}~{V0A}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjded}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((p~V1q){\unicdcdcdid}(p~V2r))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}((\preoh (\\
p~V2r)){\unicdcdcdid}(\preoh (p~V1q)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V0p)))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}\\
((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpffgbh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdcdcdhd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V2r)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}\\
(\preoh (p~V0p))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}(\preoh (p~V0p)))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfdgpgogkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdcdcdid}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(\preoh (p~V1q))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}\\
((p~V1q){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfegjgdhkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  (((p~V0p){\unicdbdjded}(\preoh (p~V1q))){\unicdbdjded}(((p~V0p){\unicdcdcdid}\\
(p~V1q)){\unicdcdcdhd}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfogfghg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((ap~(c\_2Ewords\_2Edimword~A\_27a)~\\
(c\_2Ebool\_2Ethe\_\_value~A\_27a))=(ap~(ap~c\_2Earithmetic\_2EEXP~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))~\\
(ap~(c\_2Efcp\_2Edimindex~A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a))))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfeegjgnghhpgchegpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0w \in (ty\_2Efcp\_2Ecart~\\
2~A\_27a) .  ({\unicdcdaddd} V1n \in ty\_2Enum\_2Enum .  ((V0w=(ap~(\mname{c\_2Ewords\_2En2w}~\\
A\_27a)~V1n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1n)~(ap~(c\_2Ewords\_2Edimword~\\
A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a)))))))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfechbgoghgfgegpfpfhhpgchegpfpfogdgigpgehpgngjh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((ap~(ap~(\mname{c\_2Ewords\_2Eword\_\_add}~\\
A\_27a)~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~V0m))~(ap~(\mname{c\_2Ewords\_2En2w}~\\
A\_27a)~V1n))=(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfehhpgchegpfpfbgegegpfpfogcdhh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\\
(ap~(\mname{c\_2Ewords\_2Eword\_\_1comp}~A\_27a)~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~\\
V0n))=(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~(ap~c\_2Earithmetic\_2E\_2D~(ap~(c\_2Ewords\_2Edimword~A\_27a)~\\
(c\_2Ebool\_2Ethe\_\_value~A\_27a)))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~(ap~(ap~\\
c\_2Earithmetic\_2EMOD~V0n)~(ap~(c\_2Ewords\_2Edimword~A\_27a)~(c\_2Ebool\_2Ethe\_\_value~\\
A\_27a)))))))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfehhpgchegpfpfbddgpgngahpfpfogcdhh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\\
(ap~(\mname{c\_2Ewords\_2Eword\_\_2comp}~A\_27a)~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~\\
V0n))=(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~(c\_2Ewords\_2Edimword~A\_27a)~(c\_2Ebool\_2Ethe\_\_value~A\_27a)))~\\
(ap~(ap~c\_2Earithmetic\_2EMOD~V0n)~(ap~(c\_2Ewords\_2Edimword~A\_27a)~\\
(c\_2Ebool\_2Ethe\_\_value~A\_27a)))))))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfehhpgchegpfpfcddgpgngahpfpfogcdhh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfehhpgchegdhpfcdfehfpecfeepfpfoefehe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0w \in (ty\_2Efcp\_2Ecart~\\
2~A\_27a) .  ((ap~(\mname{c\_2Ewords\_2Eword\_\_2comp}~A\_27a)~V0w)=(ap~(ap~(\\
\mname{c\_2Ewords\_2Eword\_\_add}~A\_27a)~(ap~(\mname{c\_2Ewords\_2Eword\_\_1comp}~A\_27a)~\\
V0w))~(ap~(\mname{c\_2Ewords\_2En2w}~A\_27a)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))))\end{array}$$
\end{theorem}
\end{document}
