\documentclass{article}
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\begin{document}
\title{thm\_2Ebool\_2Eliteral\_\_case\_\_RATOR (TMLuPNx6QjXURwj4KkEHG2iwRRq3Dm8D9oT)}
\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:dgpfcdfecgpgpgmgpfcdfemgjgehfgchbgmgpfpfdgbgdhfg}
We define $\mname{c\_2Ebool\_2Eliteral\_\_case}$ 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}
\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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1x \in A\_27a .  ((ap~(ap~(\mname{c\_2Ebool\_2Eliteral\_\_case}~\\
A\_27a~A\_27b)~V0f)~V1x)=(ap~V0f~V1x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemgjgehfgchbgmgpfpfdgbgdhfgpfpfefiene}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfecgpgpgmgpfcdfemgjgehfgchbgmgpfpfdgbgdhfgpfpfcfbeefpecf}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0N \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1M \in \\
A\_27a .  ({\unicdcdadad} V2b \in A\_27b .  ((ap~(ap~(ap~(\mname{c\_2Ebool\_2Eliteral\_\_case}~\\
A\_27a~(A\_27c^{{A\_27b}}))~({\uniddcgcg} V3x \in A\_27a .  (ap~V0N~V3x)))~V1M)~V2b)=(\\
ap~(ap~(\mname{c\_2Ebool\_2Eliteral\_\_case}~A\_27a~A\_27c)~({\uniddcgcg} V4x \in A\_27a .  \\
(ap~(ap~V0N~V4x)~V2b)))~V1M)))))\end{array}$$
\end{theorem}
\end{document}
