\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\_2Epre\_\_aplextoto (TMXN1hNugraMtWu79KDU7gdy29oRJ8VQA4F)}
\maketitle
Let $ ty\_2EternaryComparisons\_2Eordering : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ogfgpfehjhpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghg}
\end{equation}
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}
Let $ c\_2Etoto\_2ETO : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Etoto\_2ETO~A\_27a \in ((ty\_2Etoto\_2Etoto~\\
A\_27a)^{{({{({{ty\_2EternaryComparisons\_2Eordering}^{{A\_27a}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehpgehpgpfcdfeefpe}
\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:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2EternaryComparisons\_2Eordering2num : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EternaryComparisons\_2Eordering2num \in (ty\_2Enum\_2Enum^{{ty\_2EternaryComparisons\_2Eordering}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghgcdogfhng}
\end{equation}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\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:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\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}
\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}
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: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}
\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: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: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}
\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: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: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: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:dgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghgpfpfdebedffe}
We define $\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {ty\_2EternaryComparisons\_2Eordering} .  {{\uniddcgcg} V1v0 \in {A\_27a} .  {{\uniddcgcg} V2v1 \in {A\_27a} .  {{\uniddcgcg} V3v2 \in {A\_27a} .  {({{{ap}~{({{\uniddcgcg} V4m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V4m}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{V1v0}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V4m}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{V2v1}})}}~{V3v2}})}})}})}}~{({{{ap}~{c\_2EternaryComparisons\_2Eordering2num}}~{V0x}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeehpgehpgpfcdfedfehchpgoghgmejgogfgbgchpechegfgchpfpfpgggpfpfefpe}
We define $\mname{c\_2Etoto\_2EStrongLinearOrder\_\_of\_\_TO}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0c \in {({{({{ty\_2EternaryComparisons\_2Eordering}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}}~{2}})}}~{({{{ap}~{({{{ap}~{V0c}}~{V1x}})}}~{V2y}})}})}}~{\mname{c\_2Ebool\_2ET}}})}}~{\mname{c\_2Ebool\_2EF}}})}}~{\mname{c\_2Ebool\_2EF}}})}}}}}$.
\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}
Let $ ty\_2Epair\_2Eprod : {{\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\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Epair\_2ESND : {{\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\_2Epair\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\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\_2Epair\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfeffoedeffcfcfjf}
We define $\mname{c\_2Epair\_2EUNCURRY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1v \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{V0f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfemefeif}
We define $\mname{c\_2Epair\_2ELEX}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R1 \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1R2 \in {({{({{2}^{{A\_27b}}})}^{{A\_27b}}})} .  {({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27b}}~{({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}})}})}}~{({{\uniddcgcg} V2s \in {A\_27a} .  {({{\uniddcgcg} V3t \in {A\_27b} .  {({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27b}}~{2}})}}~{({{\uniddcgcg} V4u \in {A\_27a} .  {({{\uniddcgcg} V5v \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{V0R1}}~{V2s}})}}~{V4u}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2s}})}}~{V4u}})}})}}~{({{{ap}~{({{{ap}~{V1R2}}~{V3t}})}}~{V5v}})}})}})}})}})}})}})}})}})}}}}}$.
\end{definition}
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\_2ELESS : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2EternaryComparisons\_2ELESS} \in {ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfemefedfdf}
\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:dgpfcdfeehpgehpgpfcdfeefpepfpfpgggpfpfmejgogfgbgchpechegfgch}
We define $\mname{c\_2Etoto\_2ETO\_\_of\_\_LinearOrder}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{c\_2EternaryComparisons\_2EEQUAL}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{V0r}}~{V1x}})}}~{V2y}})}})}}~{c\_2EternaryComparisons\_2ELESS}})}}~{c\_2EternaryComparisons\_2EGREATER}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeehpgehpgpfcdfemgfgihefpe}
We define $\mname{c\_2Etoto\_2ElexTO}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{ty\_2EternaryComparisons\_2Eordering}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1V \in {({{({{ty\_2EternaryComparisons\_2Eordering}^{{A\_27b}}})}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Etoto\_2ETO\_\_of\_\_LinearOrder}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2ELEX}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Etoto\_2EStrongLinearOrder\_\_of\_\_TO}}~{A\_27a}})}}~{V0R}})}})}}~{({{{ap}~{({{\mname{c\_2Etoto\_2EStrongLinearOrder\_\_of\_\_TO}}~{A\_27b}})}}~{V1V}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeehpgehpgpfcdfemgfgihehpgehpg}
We define $\mname{c\_2Etoto\_2Elextoto}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0c \in {({{ty\_2Etoto\_2Etoto}~{A\_27a}})} .  {{\uniddcgcg} V1v \in {({{ty\_2Etoto\_2Etoto}~{A\_27b}})} .  {({{{ap}~{({{c\_2Etoto\_2ETO}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Etoto\_2ElexTO}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{c\_2Etoto\_2Eapto}~{A\_27a}})}}~{V0c}})}})}}~{({{{ap}~{({{c\_2Etoto\_2Eapto}~{A\_27b}})}}~{V1v}})}})}})}}}}}$.
\end{definition}
\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} 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} 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}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in ((ty\_2EternaryComparisons\_2Eordering^{{A\_27a}})^{{A\_27a}}) .  \\
((p~(ap~(\mname{c\_2Etoto\_2ETotOrd}~A\_27a)~V0r)){\unicdbdjdcd}((ap~(c\_2Etoto\_2Eapto~\\
A\_27a)~(ap~(c\_2Etoto\_2ETO~A\_27a)~V0r))=V0r)))\end{array}
\label{hyp:ehigngpfcdfeehpgehpgpfcdfeefpepfpfbgahehpgpfpfefpepfpfjeneaf}
\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} V0R \in ((ty\_2EternaryComparisons\_2Eordering^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V1V \in ((ty\_2EternaryComparisons\_2Eordering^{{A\_27b}})^{{A\_27b}}) .  \\
(((p~(ap~(\mname{c\_2Etoto\_2ETotOrd}~A\_27a)~V0R)){\unicdcdcdhd}(p~(ap~(\mname{c\_2Etoto\_2ETotOrd}~\\
A\_27b)~V1V))){\unicdbdjdcd}({\unicdcdadad} V2x \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  (\\
{\unicdcdadad} V3y \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ((ap~(ap~(ap~(ap~(\mname{c\_2Etoto\_2ElexTO}~\\
A\_27a~A\_27b)~V0R)~V1V)~V2x)~V3y)=(ap~(ap~(ap~(ap~(\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}~\\
ty\_2EternaryComparisons\_2Eordering)~(ap~(ap~V0R~(ap~(c\_2Epair\_2EFST~\\
A\_27a~A\_27b)~V2x))~(ap~(c\_2Epair\_2EFST~A\_27a~A\_27b)~V3y)))~c\_2EternaryComparisons\_2ELESS)~\\
(ap~(ap~V1V~(ap~(c\_2Epair\_2ESND~A\_27a~A\_27b)~V2x))~(ap~(c\_2Epair\_2ESND~\\
A\_27a~A\_27b)~V3y)))~c\_2EternaryComparisons\_2EGREATER)))))))\end{array}
\label{hyp:ehigngpfcdfeehpgehpgpfcdfemgfgihefpepfpfehigng}
\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} V0R \in ((ty\_2EternaryComparisons\_2Eordering^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V1V \in ((ty\_2EternaryComparisons\_2Eordering^{{A\_27b}})^{{A\_27b}}) .  \\
(((p~(ap~(\mname{c\_2Etoto\_2ETotOrd}~A\_27a)~V0R)){\unicdcdcdhd}(p~(ap~(\mname{c\_2Etoto\_2ETotOrd}~\\
A\_27b)~V1V))){\unicdbdjdcd}(p~(ap~(\mname{c\_2Etoto\_2ETotOrd}~(ty\_2Epair\_2Eprod~A\_27a~\\
A\_27b))~(ap~(ap~(\mname{c\_2Etoto\_2ElexTO}~A\_27a~A\_27b)~V0R)~V1V))))))\end{array}
\label{hyp:ehigngpfcdfeehpgehpgpfcdfeefpepfpfmgfgihefpe}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeehpgehpgpfcdfeahchfgpfpfbgahmgfgihehpgehpg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0c \in (ty\_2Etoto\_2Etoto~A\_27a) .  ({\unicdcdadad} V1v \in (ty\_2Etoto\_2Etoto~\\
A\_27b) .  ({\unicdcdadad} V2x \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ({\unicdcdadad} V3y \in \\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ((ap~(ap~(ap~(c\_2Etoto\_2Eapto~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~(ap~(ap~(\mname{c\_2Etoto\_2Elextoto}~A\_27a~\\
A\_27b)~V0c)~V1v))~V2x)~V3y)=(ap~(ap~(ap~(ap~(\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}~\\
ty\_2EternaryComparisons\_2Eordering)~(ap~(ap~(ap~(c\_2Etoto\_2Eapto~\\
A\_27a)~V0c)~(ap~(c\_2Epair\_2EFST~A\_27a~A\_27b)~V2x))~(ap~(c\_2Epair\_2EFST~\\
A\_27a~A\_27b)~V3y)))~c\_2EternaryComparisons\_2ELESS)~(ap~(ap~(ap~\\
(c\_2Etoto\_2Eapto~A\_27b)~V1v)~(ap~(c\_2Epair\_2ESND~A\_27a~A\_27b)~\\
V2x))~(ap~(c\_2Epair\_2ESND~A\_27a~A\_27b)~V3y)))~c\_2EternaryComparisons\_2EGREATER))))))\end{array}$$
\end{theorem}
\end{document}
