\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\_2Ereal\_2EREAL\_\_SUP\_\_LE (TMK5HNjsvHNZkYL3MQnfDd2npgVJQzBTd2i)}
\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}
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 $ ty\_2Erealax\_2Ereal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erealax\_2Ereal}}\label{hyp:ogfgpfehjhpfcdfechfgbgmgbgihpfcdfechfgbgmg}
\end{equation}
Let $ c\_2Erealax\_2Ereal\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Ereal\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{ty\_2Erealax\_2Ereal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfcffeafpfpfdemebedfdf}
\end{equation}
\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:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfcffeaf}
We define $\mname{c\_2Erealax\_2Ereal\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})}}~{({{{ap}~{c\_2Erealax\_2Ereal\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
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}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfmgeh}
We define $\mname{c\_2Erealax\_2Ereal\_\_lt}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_lt}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}}}$.
\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: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:dgpfcdfechfgbgmgpfcdfedhfhah}
We define $\mname{c\_2Ereal\_2Esup}$ to be ${{\uniddcgcg} V0P \in {({{2}^{{ty\_2Erealax\_2Ereal}}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V1s \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V2y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V3x \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V0P}}~{V3x}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{V2y}})}}~{V3x}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{V2y}})}}~{V1s}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfechfgbgmgpfpfmgehfg}
We define $\mname{c\_2Ereal\_2Ereal\_\_lte}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{V1y}})}}~{V0x}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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} V0P \in (2^{{ty\_2Erealax\_2Ereal}}) .  ((({\unicdcdaddd} V1x \in ty\_2Erealax\_2Ereal .  \\
(p~(ap~V0P~V1x))){\unicdcdcdhd}({\unicdcdaddd} V2z \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3x \in \\
ty\_2Erealax\_2Ereal .  ((p~(ap~V0P~V3x)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
V3x)~V2z)))))){\unicdbdjdcd}({\unicdcdadad} V4y \in ty\_2Erealax\_2Ereal .  (({\unicdcdaddd} V5x \in \\
ty\_2Erealax\_2Ereal .  ((p~(ap~V0P~V5x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
V4y)~V5x)))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V4y)~(ap~\mname{c\_2Ereal\_2Esup}~\\
V0P)))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfdfffaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0P \in (2^{{ty\_2Erealax\_2Ereal}}) .  ((({\unicdcdaddd} V1x \in ty\_2Erealax\_2Ereal .  \\
(p~(ap~V0P~V1x))){\unicdcdcdhd}({\unicdcdaddd} V2z \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3x \in \\
ty\_2Erealax\_2Ereal .  ((p~(ap~V0P~V3x)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V3x)~V2z)))))){\unicdbdjded}(({\unicdcdaddd} V4x \in ty\_2Erealax\_2Ereal .  (p~(ap~V0P~V4x))){\unicdcdcdhd}\\
({\unicdcdaddd} V5z \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V6x \in ty\_2Erealax\_2Ereal .  \\
((p~(ap~V0P~V6x)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V6x)~V5z))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfedffeefpelepfpfmefepfpfmeef}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfdfffafpfpfmefe}
$$\begin{array}{c}({\unicdcdadad} V0P \in (2^{{ty\_2Erealax\_2Ereal}}) .  ((({\unicdcdaddd} V1x \in ty\_2Erealax\_2Ereal .  \\
(p~(ap~V0P~V1x))){\unicdcdcdhd}({\unicdcdaddd} V2z \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3x \in \\
ty\_2Erealax\_2Ereal .  ((p~(ap~V0P~V3x)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V3x)~V2z)))))){\unicdbdjdcd}({\unicdcdadad} V4y \in ty\_2Erealax\_2Ereal .  (({\unicdcdaddd} V5x \in \\
ty\_2Erealax\_2Ereal .  ((p~(ap~V0P~V5x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
V4y)~V5x)))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V4y)~(ap~\mname{c\_2Ereal\_2Esup}~\\
V0P)))))))\end{array}$$
\end{theorem}
\end{document}
