\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\_2Ewellorder\_2EelsOf\_\_ADD1 (TMQv4nXLLrTnSwtf7zHqUeM4g2xfSvzqfdZ)}
\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:dgpfcdfecgpgpgmgpfcdfecepeffoeeefeee}
We define $\mname{c\_2Ebool\_2EBOUNDED}$ to be ${({{\uniddcgcg} V0v \in {2} .  {\mname{c\_2Ebool\_2ET}}})}$.
\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:dgpfcdfengbgchlgfgchpfcdfedepgoghg}
We define $\mname{c\_2Emarker\_2ECong}$ to be ${{\uniddcgcg} V0x \in {2} .  {V0x}}$.
\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: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: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: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\_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\_2Epair\_2EABS\_\_prod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\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}
Let $ c\_2Epred\_\_set\_2EGSPEC : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b \in ((2^{{A\_27a}})^{{({{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{2}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffede}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechbgoghgfg}
We define $\mname{c\_2Eset\_\_relation\_2Erange}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V1y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V1y}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V2x \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{A\_27a}})}}~{V2x}})}}~{V1y}})}})}}~{V0r}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeegpgngbgjgog}
We define $\mname{c\_2Eset\_\_relation\_2Edomain}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V1x}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V2y \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27b}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}})}})}})}})}}}}$.
\end{definition}
Let $ ty\_2Ewellorder\_2Ewellorder : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ewellorder\_2Ewellorder}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgch}
\end{equation}
Let $ c\_2Ewellorder\_2Ewellorder\_\_ABS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewellorder\_2Ewellorder\_\_ABS~\\
A\_27a \in ((ty\_2Ewellorder\_2Ewellorder~A\_27a)^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgchpfpfbecedf}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfefeneafefjf}
We define $\mname{c\_2Epred\_\_set\_2EEMPTY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2EF}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoedffecfef}
We define $\mname{c\_2Epred\_\_set\_2EINSERT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2y}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2y}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2y}})}}~{V1s}})}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Ewellorder\_2Ewellorder\_\_REP : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ewellorder\_2Ewellorder\_\_REP~\\
A\_27a \in ((2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})^{{({{ty\_2Ewellorder\_2Ewellorder}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgchpfpfcffeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejepeoe}
We define $\mname{c\_2Epred\_\_set\_2EUNION}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhfgmgmgpgchegfgchpfcdfefgmgdhpegg}
We define $\mname{c\_2Ewellorder\_2EelsOf}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{ty\_2Ewellorder\_2Ewellorder}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_REP}~{A\_27a}})}}~{V0w}})}})}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_REP}~{A\_27a}})}}~{V0w}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhfgmgmgpgchegfgchpfcdfebeeeeebd}
We define $\mname{c\_2Ewellorder\_2EADD1}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {{\uniddcgcg} V1w \in {({{ty\_2Ewellorder\_2Ewellorder}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{ty\_2Ewellorder\_2Ewellorder}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V0e}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27a}})}}~{V1w}})}})}})}}~{V1w}})}}~{({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_ABS}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_REP}~{A\_27a}})}}~{V1w}})}})}}~{({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V0e}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27a}})}}~{V1w}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINSERT}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V0e}})}}~{V0e}})}})}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgggmgfgihjgghfg}
We define $\mname{c\_2Eset\_\_relation\_2Ereflexive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V2x}})}})}}~{V0r}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfebgogehjgdhjhng}
We define $\mname{c\_2Eset\_\_relation\_2Eantisym}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2y}})}}~{V1x}})}})}}~{V0r}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeehchbgogdhjgehjgghfg}
We define $\mname{c\_2Eset\_\_relation\_2Etransitive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2y}})}}~{V3z}})}})}}~{V0r}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V3z}})}})}}~{V0r}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfedfffcedffeef}
We define $\mname{c\_2Epred\_\_set\_2ESUBSET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfemgjgogfgbgchpfpfpgchegfgch}
We define $\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0r}})}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0r}})}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Etransitive}}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Eantisym}}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3y}})}}~{V1s}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V3y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V3y}})}}~{V2x}})}})}}~{V0r}})}})}})}})}})}})}})}})}})}})}})}}}}$.
\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 $ c\_2Epair\_2ESND : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfeffoedeffcfcfjf}
We define $\mname{c\_2Epair\_2EUNCURRY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1v \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{V0f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfedhehchjgdgeh}
We define $\mname{c\_2Eset\_\_relation\_2Estrict}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27a}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgggpgfhogegfgeg}
We define $\mname{c\_2Ewellorder\_2Ewellfounded}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2w \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2w}})}}~{V1s}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3min \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3min}})}}~{V1s}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4w \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V4w}})}}~{V3min}})}})}}~{V0R}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V4w}})}}~{V1s}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgch}
We define $\mname{c\_2Ewellorder\_2Ewellorder}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2Ewellfounded}}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Estrict}}~{A\_27a}})}}~{V0R}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}}~{A\_27a}})}}~{V0R}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Ereflexive}}~{A\_27a}})}}~{V0R}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}})}})}})}})}}}$.
\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}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({{p}~{V0t}})}{\unicdcdcdid}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifdemeffeefeeepfpfnejeeeeemefe}
\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{\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} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{({{\unicdcdadad} V0x \in {A\_27a} .  {({{V0x}={V0x}})}})}}}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfcffegeme}
\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} 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} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  (((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1){\unicdcdcdhd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~\\
V0t1)~V1t2)=V1t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
(2^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdcdcdhd}(p~(ap~V1Q~V2x)))){\unicdbdjded}\\
(({\unicdcdadad} V3x \in A\_27a .  (p~(ap~V0P~V3x))){\unicdcdcdhd}({\unicdcdadad} V4x \in A\_27a .  (p~(\\
ap~V1Q~V4x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfbeoeeepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  ((((p~V1B){\unicdcdcdhd}\\
(p~V2C)){\unicdcdcdid}(p~V0A)){\unicdbdjded}(((p~V1B){\unicdcdcdid}(p~V0A)){\unicdcdcdhd}((p~V2C){\unicdcdcdid}(p~V0A)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfpecfpfpfpegffecfpfpfbeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  (({\unicdcdaddd} V2x \in A\_27a .  ((V2x=V1a){\unicdcdcdhd}(p~(ap~V0P~V2x)))){\unicdbdjded}(p~(\\
ap~V0P~V1a)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0v \in {2} .  {({{({{p}~{({{{ap}~{\mname{c\_2Ebool\_2EBOUNDED}}}~{V0v}})}})}{\unicdbdjded}{True}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecepeffoeeefeeepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeafbejecfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedemepedffeeepfpfafbejecfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ((V0s=V1t){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V0s)){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfefeifeffeoedfjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in ((ty\_2Epair\_2Eprod~A\_27a~2)^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1v)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b)~V0f))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~2)~V1v)~\mname{c\_2Ebool\_2ET})=(ap~V0f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffedejegejedebeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (\preoh (p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfeoepeefpfpfjeoepfpffeneafefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V2x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejepeoe}
\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} V2s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V0x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V1y)~V2s))){\unicdbdjded}((V0x=\\
V1y){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~V2s)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfjeoedffecfef}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0t}})}}})}}})}{\unicdbdjded}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeoepeefpfpfoepeef}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{p}~{V0A}})}{\unicdbdjdcd}{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfpfpfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
(((p~V0A){\unicdbdjdcd}False){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((\preoh (p~V0A)){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
((p~V0A){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemedd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{({{({{({{p}~{V0A}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjded}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((p~V1q){\unicdcdcdid}(p~V2r))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}((\preoh (\\
p~V2r)){\unicdcdcdid}(\preoh (p~V1q)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V0p)))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}\\
((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpffgbh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdcdcdhd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V2r)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}\\
(\preoh (p~V0p))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}(\preoh (p~V0p)))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfdgpgogkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdcdcdid}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(\preoh (p~V1q))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}\\
((p~V1q){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfegjgdhkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjdcd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(p~V1q)){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}((\\
\preoh (p~V1q)){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfjgngah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  (((p~V0p){\unicdbdjded}(\preoh (p~V1q))){\unicdbdjded}(((p~V0p){\unicdcdcdid}\\
(p~V1q)){\unicdcdcdhd}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfogfghg}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdbdjdcd}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({{p}~{V0p}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogjgbd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdbdjdcd}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V1q}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogjgcd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdcdcdid}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V0p}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogpgbd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdcdcdid}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V1q}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogpgcd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0p}})}}})}}})}{\unicdbdjdcd}{({{p}~{V0p}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogog}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(ap~(\mname{c\_2Eset\_\_relation\_2Edomain}~\\
A\_27a~A\_27b)~V1r))){\unicdbdjded}({\unicdcdaddd} V2y \in A\_27b .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~\\
A\_27b)~V0x)~V2y))~V1r))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfejgogpfpfegpgngbgjgog}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0y \in A\_27a .  ({\unicdcdadad} V1r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}}) .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0y)~(ap~(\mname{c\_2Eset\_\_relation\_2Erange}~\\
A\_27a~A\_27b)~V1r))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27b .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
(ty\_2Epair\_2Eprod~A\_27b~A\_27a))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27b~\\
A\_27a)~V2x)~V0y))~V1r))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfejgogpfpfchbgoghgfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0a \in (ty\_2Ewellorder\_2Ewellorder~\\
A\_27a) .  ((ap~(c\_2Ewellorder\_2Ewellorder\_\_ABS~A\_27a)~(ap~(c\_2Ewellorder\_2Ewellorder\_\_REP~\\
A\_27a)~V0a))=V0a)){\unicdcdcdhd}({\unicdcdadad} V1r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((p~(ap~(\mname{c\_2Ewellorder\_2Ewellorder}~A\_27a)~V1r)){\unicdbdjded}((ap~(c\_2Ewellorder\_2Ewellorder\_\_REP~\\
A\_27a)~(ap~(c\_2Ewellorder\_2Ewellorder\_\_ABS~A\_27a)~V1r))=V1r))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgchpfpfbecedfcffeaf}
\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} V2w \in (ty\_2Ewellorder\_2Ewellorder~A\_27a) .  ((p~\\
(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Epair\_2Eprod~A\_27a~A\_27a))~(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27a)~V0x)~V1y))~(ap~(c\_2Ewellorder\_2Ewellorder\_\_REP~\\
A\_27a)~V2w))){\unicdbdjdcd}((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(ap~(\mname{c\_2Ewellorder\_2EelsOf}~\\
A\_27a)~V2w))){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1y)~(ap~(\mname{c\_2Ewellorder\_2EelsOf}~\\
A\_27a)~V2w))))))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfehfmefepfpffgmgdhpegg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0e \in A\_27a .  ({\unicdcdadad} V1w \in \\
(ty\_2Ewellorder\_2Ewellorder~A\_27a) .  ((\preoh (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V0e)~(ap~(\mname{c\_2Ewellorder\_2EelsOf}~A\_27a)~V1w)))){\unicdbdjdcd}(p~(ap~(\\
\mname{c\_2Ewellorder\_2Ewellorder}~A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27a))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27a))~(ap~(c\_2Ewellorder\_2Ewellorder\_\_REP~\\
A\_27a)~V1w))~(ap~(c\_2Epred\_\_set\_2EGSPEC~(ty\_2Epair\_2Eprod~A\_27a~\\
A\_27a)~A\_27a)~({\uniddcgcg} V2x \in A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27a)~2)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27a)~V2x)~V0e))~\\
(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~(ap~(\mname{c\_2Ewellorder\_2EelsOf}~\\
A\_27a)~V1w)))))))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27a))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27a)~V0e)~V0e))~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27a)))))))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgchpfpfbeeeeebd}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfefgmgdhpeggpfpfbeeeeebd}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0e \in A\_27a .  ({\unicdcdadad} V1w \in \\
(ty\_2Ewellorder\_2Ewellorder~A\_27a) .  ((ap~(\mname{c\_2Ewellorder\_2EelsOf}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Ewellorder\_2EADD1}~A\_27a)~V0e)~V1w))=(ap~(ap~\\
(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V0e)~(ap~(\mname{c\_2Ewellorder\_2EelsOf}~\\
A\_27a)~V1w)))))\end{array}$$
\end{theorem}
\end{document}
