\documentclass{article}
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\newtheorem{conjecture}{Conjecture}
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\begin{document}
\title{thm\_2Estate\_\_transformer\_2EBIND\_\_EXT (TMMjJBePHSLbo8LynTXgcGTAKDEytMrA2Zs)}
\maketitle
\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}
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: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: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:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhehbgehfgpfpfehchbgogdhggpgchngfgchpfcdfecejeoeee}
We define $\mname{c\_2Estate\_\_transformer\_2EBIND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0g \in {({{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1f \in {({{({{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27a}})}^{{A\_27a}}})}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2Eo}}~{A\_27a}}~{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27a}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27b}}~{A\_27a}}~{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27a}})}})}}~{V1f}})}})}}~{V0g}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhehbgehfgpfpfehchbgogdhggpgchngfgchpfcdfefeifef}
We define $\mname{c\_2Estate\_\_transformer\_2EEXT}$ to be ${{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} A\_27s : {\iota} .  {{\uniddcgcg} V0f \in {({{({{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27s}})}^{{A\_27s}}})}^{{A\_27b}}})} .  {{\uniddcgcg} V1m \in {({{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27s}})}^{{A\_27s}}})} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2Eo}}~{A\_27s}}~{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27s}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27s}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27b}}~{A\_27s}}~{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27s}})}})}}~{V0f}})}})}}~{V1m}})}}}}}}$.
\end{definition}
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} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedhehbgehfgpfpfehchbgogdhggpgchngfgchpfcdfecejeoeeepfpffeifef}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0m \in ((ty\_2Epair\_2Eprod~A\_27c~A\_27a)^{{A\_27a}}) .  \\
({\unicdcdadad} V1f \in (((ty\_2Epair\_2Eprod~A\_27b~A\_27a)^{{A\_27a}})^{{A\_27c}}) .  \\
((ap~(ap~(\mname{c\_2Estate\_\_transformer\_2EBIND}~A\_27a~A\_27c~A\_27b)~V0m)~\\
V1f)=(ap~(ap~(\mname{c\_2Estate\_\_transformer\_2EEXT}~A\_27c~A\_27b~A\_27a)~\\
V1f)~V0m))))\end{array}$$
\end{theorem}
\end{document}
