\documentclass{article}
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\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\begin{document}
\title{thm\_2Equotient\_2Eliteral\_\_case\_\_RSP (TMKgEWyhDWegt5PbFYvu34PeavdpRK8CCYw)}
\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: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}
\begin{definition}\label{def:dgpfcdfebhfhpgehjgfgogehpfcdfepfddeepfddeepfddeepfddfe}
We define $\mname{c\_2Equotient\_2E\_3D\_3D\_3D\_3E}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R1 \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1R2 \in {({{({{2}^{{A\_27b}}})}^{{A\_27b}}})} .  {{\uniddcgcg} V2f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V3g \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R1}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V1R2}}~{({{{ap}~{V2f}}~{V4x}})}})}}~{({{{ap}~{V3g}}~{V5y}})}})}})}})}})}})}})}}}}}}}$.
\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:dgpfcdfebhfhpgehjgfgogehpfcdfebfffpeefjefeoeef}
We define $\mname{c\_2Equotient\_2EQUOTIENT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1abs \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V2rep \in {({{A\_27a}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V3a \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V1abs}}~{({{{ap}~{V2rep}}~{V3a}})}})}})}}~{V3a}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V4a \in {A\_27b} .  {({{{ap}~{({{{ap}~{V0R}}~{({{{ap}~{V2rep}}~{V4a}})}})}}~{({{{ap}~{V2rep}}~{V4a}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5r \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6s \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V5r}})}}~{V6s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V5r}})}}~{V5r}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V6s}})}}~{V6s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V1abs}}~{V5r}})}})}}~{({{{ap}~{V1abs}}~{V6s}})}})}})}})}})}})}})}})}})}})}})}}}}}}$.
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0R1 \in (\\
(2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs1 \in (A\_27c^{{A\_27a}}) .  ({\unicdcdadad} V2rep1 \in \\
(A\_27a^{{A\_27c}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~A\_27a~A\_27c)~\\
V0R1)~V1abs1)~V2rep1)){\unicdbdjdcd}({\unicdcdadad} V3R2 \in ((2^{{A\_27b}})^{{A\_27b}}) .  ({\unicdcdadad} V4abs2 \in \\
(A\_27d^{{A\_27b}}) .  ({\unicdcdadad} V5rep2 \in (A\_27b^{{A\_27d}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27b~A\_27d)~V3R2)~V4abs2)~V5rep2)){\unicdbdjdcd}({\unicdcdadad} V6f \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V7g \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V8x \in A\_27a .  ({\unicdcdadad} V9y \in \\
A\_27a .  (((p~(ap~(ap~(ap~(ap~(\mname{c\_2Equotient\_2E\_3D\_3D\_3D\_3E}~A\_27a~\\
A\_27b)~V0R1)~V3R2)~V6f)~V7g)){\unicdcdcdhd}(p~(ap~(ap~V0R1~V8x)~V9y))){\unicdbdjdcd}(p~(ap~(\\
ap~V3R2~(ap~V6f~V8x))~(ap~V7g~V9y))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfegeffoepfpfcffemepfpfneaf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfebhfhpgehjgfgogehpfcdfemgjgehfgchbgmgpfpfdgbgdhfgpfpfcfdfaf}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0R1 \in (\\
(2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs1 \in (A\_27c^{{A\_27a}}) .  ({\unicdcdadad} V2rep1 \in \\
(A\_27a^{{A\_27c}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~A\_27a~A\_27c)~\\
V0R1)~V1abs1)~V2rep1)){\unicdbdjdcd}({\unicdcdadad} V3R2 \in ((2^{{A\_27b}})^{{A\_27b}}) .  ({\unicdcdadad} V4abs2 \in \\
(A\_27d^{{A\_27b}}) .  ({\unicdcdadad} V5rep2 \in (A\_27b^{{A\_27d}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27b~A\_27d)~V3R2)~V4abs2)~V5rep2)){\unicdbdjdcd}({\unicdcdadad} V6f \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V7g \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V8x \in A\_27a .  ({\unicdcdadad} V9y \in \\
A\_27a .  (((p~(ap~(ap~(ap~(ap~(\mname{c\_2Equotient\_2E\_3D\_3D\_3D\_3E}~A\_27a~\\
A\_27b)~V0R1)~V3R2)~V6f)~V7g)){\unicdcdcdhd}(p~(ap~(ap~V0R1~V8x)~V9y))){\unicdbdjdcd}(p~(ap~(\\
ap~V3R2~(ap~(ap~(\mname{c\_2Ebool\_2Eliteral\_\_case}~A\_27a~A\_27b)~V6f)~V8x))~\\
(ap~(ap~(\mname{c\_2Ebool\_2Eliteral\_\_case}~A\_27a~A\_27b)~V7g)~V9y))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
