\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\_2EHREAL\_\_LT\_\_LADD (TMcxVN1GDbpquRwBKPofwZPA8d92FE22amP)}
\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}
Let $ ty\_2Ehrat\_2Ehrat : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehrat\_2Ehrat}}\label{hyp:ogfgpfehjhpfcdfeigchbgehpfcdfeigchbgeh}
\end{equation}
Let $ ty\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehreal\_2Ehreal}}\label{hyp:ogfgpfehjhpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\end{equation}
Let $ c\_2Ehreal\_2Ecut : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ehreal\_2Ecut} \in {({{({{2}^{{ty\_2Ehrat\_2Ehrat}}})}^{{ty\_2Ehreal\_2Ehreal}}})}}\label{hyp:ngfgngpfdgpfcdfeigchfgbgmgpfcdfedgfheh}
\end{equation}
\begin{definition}\label{def:dgpfcdfeigchfgbgmgpfcdfeigchfgbgmgpfpfmgeh}
We define $\mname{c\_2Ehreal\_2Ehreal\_\_lt}$ to be ${{\uniddcgcg} V0X \in {ty\_2Ehreal\_2Ehreal} .  {{\uniddcgcg} V1Y \in {ty\_2Ehreal\_2Ehreal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Ehreal\_2Ehreal}})}}~{V0X}})}}~{V1Y}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V2x \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{c\_2Ehreal\_2Ecut}}~{V0X}})}}~{V2x}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ehreal\_2Ecut}}~{V1Y}})}}~{V2x}})}})}})}})}})}}}$.
\end{definition}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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\_2Ehrat\_2Ehrat\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Ehrat\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Ehrat\_2Ehrat}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeigchbgehpfpfcffeafpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeigchbgehpfcdfeigchbgehpfpfcffeaf}
We define $\mname{c\_2Ehrat\_2Ehrat\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{c\_2Ehrat\_2Ehrat\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Ehrat\_2Etrat\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Etrat\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeehchbgehpfpfbgegeg}
\end{equation}
Let $ c\_2Ehrat\_2Etrat\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Etrat\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeehchbgehpfpffgbh}
\end{equation}
Let $ c\_2Ehrat\_2Ehrat\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Ehrat\_\_ABS\_\_CLASS \in (ty\_2Ehrat\_2Ehrat^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeigchbgehpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeigchbgehpfcdfeigchbgehpfpfbecedf}
We define $\mname{c\_2Ehrat\_2Ehrat\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})} .  {({{{ap}~{c\_2Ehrat\_2Ehrat\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Ehrat\_2Etrat\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeigchbgehpfcdfeigchbgehpfpfbgegeg}
We define $\mname{c\_2Ehrat\_2Ehrat\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Ehrat\_2Ehrat} .  {{\uniddcgcg} V1T2 \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{\mname{c\_2Ehrat\_2Ehrat\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Ehrat\_2Etrat\_\_add}}~{({{{ap}~{\mname{c\_2Ehrat\_2Ehrat\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Ehrat\_2Ehrat\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
Let $ c\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ehreal\_2Ehreal} \in {({{ty\_2Ehreal\_2Ehreal}^{{({{2}^{{ty\_2Ehrat\_2Ehrat}}})}}})}}\label{hyp:ngfgngpfdgpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeigchfgbgmgpfcdfeigchfgbgmgpfpfbgegeg}
We define $\mname{c\_2Ehreal\_2Ehreal\_\_add}$ to be ${{\uniddcgcg} V0X \in {ty\_2Ehreal\_2Ehreal} .  {{\uniddcgcg} V1Y \in {ty\_2Ehreal\_2Ehreal} .  {({{{ap}~{c\_2Ehreal\_2Ehreal}}~{({{\uniddcgcg} V2w \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V3x \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V4y \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Ehrat\_2Ehrat}})}}~{V2w}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ehrat\_2Ehrat\_\_add}}}~{V3x}})}}~{V4y}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{c\_2Ehreal\_2Ecut}}~{V0X}})}}~{V3x}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ehreal\_2Ecut}}~{V1Y}})}}~{V4y}})}})}})}})}})}})}})}})}})}}}$.
\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{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\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 .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0X \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1Y \in ty\_2Ehreal\_2Ehreal .  \\
({\unicdcdadad} V2Z \in ty\_2Ehreal\_2Ehreal .  ((ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~\\
V0X)~(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~V1Y)~V2Z))=(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~\\
(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~V0X)~V1Y))~V2Z)))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfeiecffebemepfpfbeeeeepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0X \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1Y \in ty\_2Ehreal\_2Ehreal .  \\
((p~(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_lt}~V0X)~V1Y)){\unicdbdjded}({\unicdcdaddd} V2D \in ty\_2Ehreal\_2Ehreal .  \\
(V1Y=(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~V0X)~V2D))))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfeiecffebemepfpfmeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1y \in ty\_2Ehreal\_2Ehreal .  \\
({\unicdcdadad} V2z \in ty\_2Ehreal\_2Ehreal .  (((ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~\\
V0x)~V1y)=(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~V0x)~V2z)){\unicdbdjded}(V1y=V2z)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgbgihpfcdfeiecffebemepfpffebfpfpfmebeeeee}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechfgbgmgbgihpfcdfeiecffebemepfpfmeefpfpfmebeeeee}
$$\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1y \in ty\_2Ehreal\_2Ehreal .  \\
({\unicdcdadad} V2z \in ty\_2Ehreal\_2Ehreal .  ((p~(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_lt}~\\
(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~V0x)~V1y))~(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_add}~\\
V0x)~V2z))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_lt}~V1y)~V2z))))))\end{array}$$
\end{theorem}
\end{document}
