\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Etc\_2EsubTC\_\_thm (TMLEVuMNSiYdS3iUN2uQ3gWwdTkAqot6pQg)}
\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}
\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:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\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:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\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:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfepe}
We define $\mname{c\_2Erelation\_2EO}$ to be ${{\uniddcgcg} A\_27g : {\iota} .  {{\uniddcgcg} A\_27h : {\iota} .  {{\uniddcgcg} A\_27k : {\iota} .  {{\uniddcgcg} V0R1 \in {({{({{2}^{{A\_27k}}})}^{{A\_27h}}})} .  {{\uniddcgcg} V1R2 \in {({{({{2}^{{A\_27h}}})}^{{A\_27g}}})} .  {{\uniddcgcg} V2x \in {A\_27g} .  {{\uniddcgcg} V3z \in {A\_27k} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27h}})}}~{({{\uniddcgcg} V4y \in {A\_27h} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V1R2}}~{V2x}})}}~{V4y}})}})}}~{({{{ap}~{({{{ap}~{V0R1}}~{V4y}})}}~{V3z}})}})}})}})}}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfecfffoejepeoe}
We define $\mname{c\_2Erelation\_2ERUNION}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R1 \in {({{({{2}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1R2 \in {({{({{2}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{V0R1}}~{V2x}})}}~{V3y}})}})}}~{({{{ap}~{({{{ap}~{V1R2}}~{V2x}})}}~{V3y}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfejeoe}
We define $\mname{c\_2Ebool\_2EIN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1f \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V1f}}~{V0x}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeehdgpfcdfepffdfepfhdde}
We define $\mname{c\_2Etc\_2E\_5E\_7C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2a \in {A\_27a} .  {{\uniddcgcg} V3b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2a}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V2a}})}}~{V3b}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeehdgpfcdfepfhddepffdfe}
We define $\mname{c\_2Etc\_2E\_7C\_5E}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2a \in {A\_27a} .  {{\uniddcgcg} V3b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3b}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V2a}})}}~{V3b}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeehdgpfcdfepffdfepfhddepffdfe}
We define $\mname{c\_2Etc\_2E\_5E\_7C\_5E}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Etc\_2E\_7C\_5E}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Etc\_2E\_5E\_7C}}~{A\_27a}})}}~{V0R}})}}~{V1s}})}})}}~{V1s}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfecfefde}
We define $\mname{c\_2Erelation\_2ERTC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3P \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V3P}}~{V4x}})}}~{V4x}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V5x}})}}~{V6y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V6y}})}}~{V7z}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V5x}})}}~{V7z}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V1a}})}}~{V2b}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeehdgpfcdfedhfhcgefde}
We define $\mname{c\_2Etc\_2EsubTC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{V0R}}~{V2x}})}}~{V3y}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V4a \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V5b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Erelation\_2ERTC}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Etc\_2E\_5E\_7C\_5E}}~{A\_27a}})}}~{V0R}})}}~{V1s}})}})}}~{V4a}})}}~{V5b}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V4a}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V5b}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V2x}})}}~{V4a}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V5b}})}}~{V3y}})}})}})}})}})}})}})}})}})}})}}}}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1R \in \\
((2^{{A\_27a}})^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3y \in A\_27a .  (\\
((p~(ap~V0P~V2x)){\unicdcdcdhd}(p~(ap~(ap~V1R~V2x)~V3y))){\unicdbdjdcd}(p~(ap~V0P~V3y))))){\unicdbdjdcd}(\\
{\unicdcdadad} V4x \in A\_27a .  ({\unicdcdadad} V5y \in A\_27a .  (((p~(ap~V0P~V4x)){\unicdcdcdhd}(p~(ap~\\
(ap~(ap~(\mname{c\_2Erelation\_2ERTC}~A\_27a)~V1R)~V4x)~V5y))){\unicdbdjdcd}(p~(ap~V0P~V5y))))))))\end{array}
\label{hyp:ehigngpfcdfechfgmgbgehjgpgogpfcdfecfefdepfpfmgjgggehdhpfpfjgogghbgchjgbgogehdh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeehdgpfcdfedhfhcgefdepfpfehigng}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0R \in ((2^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ((ap~(ap~(\mname{c\_2Etc\_2EsubTC}~A\_27a)~V0R)~V1s)=\\
(ap~(ap~(\mname{c\_2Erelation\_2ERUNION}~A\_27a~A\_27a)~V0R)~(ap~(ap~(\mname{c\_2Erelation\_2EO}~\\
A\_27a~A\_27a~A\_27a)~V0R)~(ap~(ap~(\mname{c\_2Erelation\_2EO}~A\_27a~A\_27a~A\_27a)~\\
(ap~(ap~(\mname{c\_2Etc\_2E\_5E\_7C}~A\_27a)~(ap~(\mname{c\_2Erelation\_2ERTC}~A\_27a)~\\
(ap~(ap~(\mname{c\_2Etc\_2E\_5E\_7C\_5E}~A\_27a)~V0R)~V1s)))~V1s))~V0R))))))\end{array}$$
\end{theorem}
\end{document}
