\documentclass{article}
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\newtheorem{conjecture}{Conjecture}
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\begin{document}
\title{thm\_2Equote\_2Ecompare\_\_list\_\_index (TMboNjPm6f2V9prqJpjkUHzBLjTjNGUuN4a)}
\maketitle
Let $ ty\_2EternaryComparisons\_2Eordering : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ogfgpfehjhpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghg}
\end{equation}
Let $ ty\_2Equote\_2Eindex : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Equote\_2Eindex}}\label{hyp:ogfgpfehjhpfcdfebhfhpgehfgpfcdfejgogegfgih}
\end{equation}
Let $ c\_2Equote\_2Eindex\_\_compare : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Equote\_2Eindex\_\_compare \in ((ty\_2EternaryComparisons\_2Eordering^{{ty\_2Equote\_2Eindex}})^{{ty\_2Equote\_2Eindex}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebhfhpgehfgpfcdfejgogegfgihpfpfdgpgngahbgchfg}
\end{equation}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ c\_2EternaryComparisons\_2Elist\_\_compare : {{\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\_2EternaryComparisons\_2Elist\_\_compare~\\
A\_27a~A\_27b \in (((ty\_2EternaryComparisons\_2Eordering^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{({{ty\_2EternaryComparisons\_2Eordering}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfemgjgdhehpfpfdgpgngahbgchfg}
\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: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: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}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0i1 \in ty\_2Equote\_2Eindex .  ({\unicdcdadad} V1i2 \in ty\_2Equote\_2Eindex .  \\
(((ap~(ap~c\_2Equote\_2Eindex\_\_compare~V0i1)~V1i2)=c\_2EternaryComparisons\_2EEQUAL){\unicdbdjded}\\
(V0i1=V1i2))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehfgpfcdfedgpgngahbgchfgpfpfjgogegfgihpfpffgbhfhbgmg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0cmp \in ((ty\_2EternaryComparisons\_2Eordering^{{A\_27a}})^{{A\_27a}}) .  \\
(({\unicdcdadad} V1x \in A\_27a .  ({\unicdcdadad} V2y \in A\_27a .  (((ap~(ap~V0cmp~V1x)~V2y)=\\
c\_2EternaryComparisons\_2EEQUAL){\unicdbdjded}(V1x=V2y)))){\unicdbdjdcd}({\unicdcdadad} V3l1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V4l2 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
(((ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_compare~A\_27a~\\
A\_27a)~V0cmp)~V3l1)~V4l2)=c\_2EternaryComparisons\_2EEQUAL){\unicdbdjded}\\
(V3l1=V4l2))))))\end{array}
\label{hyp:ehigngpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfedgpgngahbgchfgpfpffgbhfhbgmg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfebhfhpgehfgpfcdfedgpgngahbgchfgpfpfmgjgdhehpfpfjgogegfgih}
$$\begin{array}{c}({\unicdcdadad} V0l1 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V1l2 \in \\
(ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  (((ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_compare~\\
ty\_2Equote\_2Eindex~ty\_2Equote\_2Eindex)~c\_2Equote\_2Eindex\_\_compare)~\\
V0l1)~V1l2)=c\_2EternaryComparisons\_2EEQUAL){\unicdbdjded}(V0l1=V1l2))))\end{array}$$
\end{theorem}
\end{document}
