\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\_2Erealax\_2ETREAL\_\_LT\_\_WELLDEF (TMPuf3yzogpUQsEZMwsDLmoQ4qrn8C5o5zD)}
\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}
Let $ ty\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehreal\_2Ehreal}}\label{hyp:ogfgpfehjhpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\end{equation}
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\_2Erealax\_2Etreal\_\_lt : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_lt \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfmgeh}
\end{equation}
Let $ c\_2Erealax\_2Etreal\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpffgbh}
\end{equation}
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 .  ({\unicdcdadad} V1y \in \\
A\_27a .  ({\unicdcdadad} V2z \in A\_27a .  (((V0x=V1y){\unicdcdcdhd}(V1y=V2z)){\unicdbdjdcd}(V0x=V2z)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfefcfbeoedf}
\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} V0x1 \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
({\unicdcdadad} V1x2 \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
({\unicdcdadad} V2y \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
((p~(ap~(ap~c\_2Erealax\_2Etreal\_\_eq~V0x1)~V1x2)){\unicdbdjdcd}((p~(ap~(ap~c\_2Erealax\_2Etreal\_\_lt~\\
V0x1)~V2y)){\unicdbdjded}(p~(ap~(ap~c\_2Erealax\_2Etreal\_\_lt~V1x2)~V2y)))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgbgihpfcdfeefcffebemepfpfmeefpfpfhffememeeefegecf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
({\unicdcdadad} V1y1 \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
({\unicdcdadad} V2y2 \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
((p~(ap~(ap~c\_2Erealax\_2Etreal\_\_eq~V1y1)~V2y2)){\unicdbdjdcd}((p~(ap~(ap~c\_2Erealax\_2Etreal\_\_lt~\\
V0x)~V1y1)){\unicdbdjded}(p~(ap~(ap~c\_2Erealax\_2Etreal\_\_lt~V0x)~V2y2)))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgbgihpfcdfeefcffebemepfpfmeefpfpfhffememeeefegeme}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechfgbgmgbgihpfcdfeefcffebemepfpfmeefpfpfhffememeeefege}
$$\begin{array}{c}({\unicdcdadad} V0x1 \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
({\unicdcdadad} V1x2 \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
({\unicdcdadad} V2y1 \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
({\unicdcdadad} V3y2 \in (ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~ty\_2Ehreal\_2Ehreal) .  \\
(((p~(ap~(ap~c\_2Erealax\_2Etreal\_\_eq~V0x1)~V1x2)){\unicdcdcdhd}(p~(ap~(ap~c\_2Erealax\_2Etreal\_\_eq~\\
V2y1)~V3y2))){\unicdbdjdcd}((p~(ap~(ap~c\_2Erealax\_2Etreal\_\_lt~V0x1)~V2y1)){\unicdbdjded}\\
(p~(ap~(ap~c\_2Erealax\_2Etreal\_\_lt~V1x2)~V3y2))))))))\end{array}$$
\end{theorem}
\end{document}
