\documentclass{article}
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\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\begin{document}
\title{thm\_2Epair\_2ELET2\_\_RATOR (TMKZNSxBdc11jY5qeEi128426VHi76Jyd9r)}
\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:dgpfcdfecgpgpgmgpfcdfemefeef}
We define $\mname{c\_2Ebool\_2ELET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{V0f}}~{V1x}})}})}})}}}$.
\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}
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} 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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0f \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1v \in \\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ((ap~(ap~(\mname{c\_2Epair\_2EUNCURRY}~\\
A\_27a~A\_27b~A\_27c)~V0f)~V1v)=(ap~(ap~V0f~(ap~(c\_2Epair\_2EFST~A\_27a~\\
A\_27b)~V1v))~(ap~(c\_2Epair\_2ESND~A\_27a~A\_27b)~V1v)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeffoedeffcfcfjfpfpfgfbecf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeahbgjgchpfcdfemefeefcdpfpfcfbeefpecf}
$$\begin{array}{c}{\unicdcdadad} A\_27a1 .  nonempty~A\_27a1{\unicdbdjdcd}{\unicdcdadad} A\_27a2 .  nonempty~A\_27a2{\unicdbdjdcd}\\
{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  nonempty~A\_27c{\unicdbdjdcd}(\\
{\unicdcdadad} V0M \in (ty\_2Epair\_2Eprod~A\_27a1~A\_27a2) .  ({\unicdcdadad} V1N \in (\\
((A\_27c^{{A\_27b}})^{{A\_27a2}})^{{A\_27a1}}) .  ({\unicdcdadad} V2b \in A\_27b .  ((ap~(ap~(ap~\\
(\mname{c\_2Ebool\_2ELET}~(ty\_2Epair\_2Eprod~A\_27a1~A\_27a2)~(A\_27c^{{A\_27b}}))~\\
(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a1~A\_27a2~(A\_27c^{{A\_27b}}))~({\uniddcgcg} V3x \in \\
A\_27a1 .  ({\uniddcgcg} V4y \in A\_27a2 .  (ap~(ap~V1N~V3x)~V4y)))))~V0M)~V2b)=(ap~\\
(ap~(\mname{c\_2Ebool\_2ELET}~(ty\_2Epair\_2Eprod~A\_27a1~A\_27a2)~A\_27c)~(\\
ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a1~A\_27a2~A\_27c)~({\uniddcgcg} V5x \in A\_27a1 .  \\
({\uniddcgcg} V6y \in A\_27a2 .  (ap~(ap~(ap~V1N~V5x)~V6y)~V2b)))))~V0M)))))\end{array}$$
\end{theorem}
\end{document}
