\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\_2Etoto\_2Etoto\_\_thm (TMNUwdBAqofS9QhCz54pmD333crewKz2gEo)}
\maketitle
\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}
Let $ ty\_2EternaryComparisons\_2Eordering : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ogfgpfehjhpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghg}
\end{equation}
Let $ c\_2EternaryComparisons\_2ELESS : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2EternaryComparisons\_2ELESS} \in {ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfemefedfdf}
\end{equation}
Let $ c\_2EternaryComparisons\_2EGREATER : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EternaryComparisons\_2EGREATER \in ty\_2EternaryComparisons\_2Eordering\end{array}
\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfehecffebeeffecf}
\end{equation}
Let $ c\_2EternaryComparisons\_2EEQUAL : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2EternaryComparisons\_2EEQUAL} \in {ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfefebfffbeme}
\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}
\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: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 $ ty\_2Etoto\_2Etoto : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Etoto\_2Etoto}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeehpgehpgpfcdfeehpgehpg}
\end{equation}
Let $ c\_2Etoto\_2Eapto : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Etoto\_2Eapto~A\_27a \in (((ty\_2EternaryComparisons\_2Eordering^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Etoto\_2Etoto}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehpgehpgpfcdfebgahehpg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeehpgehpgpfcdfeefpgehpecheg}
We define $\mname{c\_2Etoto\_2ETotOrd}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0c \in {({{({{ty\_2EternaryComparisons\_2Eordering}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{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}}~{2}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{V0c}}~{V1x}})}}~{V2y}})}})}}~{c\_2EternaryComparisons\_2EEQUAL}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{V0c}}~{V3x}})}}~{V4y}})}})}}~{c\_2EternaryComparisons\_2EGREATER}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{V0c}}~{V4y}})}}~{V3x}})}})}}~{c\_2EternaryComparisons\_2ELESS}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{V0c}}~{V5x}})}}~{V6y}})}})}}~{c\_2EternaryComparisons\_2ELESS}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{V0c}}~{V6y}})}}~{V7z}})}})}}~{c\_2EternaryComparisons\_2ELESS}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{V0c}}~{V5x}})}}~{V7z}})}})}}~{c\_2EternaryComparisons\_2ELESS}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0c \in (ty\_2Etoto\_2Etoto~\\
A\_27a) .  (p~(ap~(\mname{c\_2Etoto\_2ETotOrd}~A\_27a)~(ap~(c\_2Etoto\_2Eapto~\\
A\_27a)~V0c))))\end{array}
\label{hyp:ehigngpfcdfeehpgehpgpfcdfeefpgehpechegpfpfbgahehpg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeehpgehpgpfcdfeehpgehpgpfpfehigng}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0c \in (ty\_2Etoto\_2Etoto~\\
A\_27a) .  (({\unicdcdadad} V1x \in A\_27a .  ({\unicdcdadad} V2y \in A\_27a .  (((ap~(ap~(ap~\\
(c\_2Etoto\_2Eapto~A\_27a)~V0c)~V1x)~V2y)=c\_2EternaryComparisons\_2EEQUAL){\unicdbdjded}\\
(V1x=V2y)))){\unicdcdcdhd}(({\unicdcdadad} V3x \in A\_27a .  ({\unicdcdadad} V4y \in A\_27a .  (((ap~\\
(ap~(ap~(c\_2Etoto\_2Eapto~A\_27a)~V0c)~V3x)~V4y)=c\_2EternaryComparisons\_2EGREATER){\unicdbdjded}\\
((ap~(ap~(ap~(c\_2Etoto\_2Eapto~A\_27a)~V0c)~V4y)~V3x)=c\_2EternaryComparisons\_2ELESS)))){\unicdcdcdhd}\\
({\unicdcdadad} V5x \in A\_27a .  ({\unicdcdadad} V6y \in A\_27a .  ({\unicdcdadad} V7z \in A\_27a .  (\\
(((ap~(ap~(ap~(c\_2Etoto\_2Eapto~A\_27a)~V0c)~V5x)~V6y)=c\_2EternaryComparisons\_2ELESS){\unicdcdcdhd}\\
((ap~(ap~(ap~(c\_2Etoto\_2Eapto~A\_27a)~V0c)~V6y)~V7z)=c\_2EternaryComparisons\_2ELESS)){\unicdbdjdcd}\\
((ap~(ap~(ap~(c\_2Etoto\_2Eapto~A\_27a)~V0c)~V5x)~V7z)=c\_2EternaryComparisons\_2ELESS))))))))\end{array}$$
\end{theorem}
\end{document}
