\documentclass{article}
\newtheorem{axiom}{Axiom}
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\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}
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\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Eordinal\_2Eordlt\_\_CANCEL (TMTMy6XE3hM2gHc6iLuoXcuADfE8VDF24eA)}
\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}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ ty\_2Esum\_2Esum : {{\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\_2Esum\_2Esum~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfedhfhngpfcdfedhfhng}
\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\_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\_\_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: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 $ 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\_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}
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: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: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: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}
\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: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:dgpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchjgdhpg}
We define $\mname{c\_2Ewellorder\_2Eorderiso}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0w1 \in {({{ty\_2Ewellorder\_2Ewellorder}~{A\_27a}})} .  {{\uniddcgcg} V1w2 \in {({{ty\_2Ewellorder\_2Ewellorder}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3x}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27a}})}}~{V0w1}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{({{{ap}~{V2f}}~{V3x}})}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27b}})}}~{V1w2}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x1 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5x2 \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}})}}~{V4x1}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27a}})}}~{V0w1}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V5x2}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27a}})}}~{V0w1}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V2f}}~{V4x1}})}})}}~{({{{ap}~{V2f}}~{V5x2}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V4x1}})}}~{V5x2}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V6y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{V6y}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27b}})}}~{V1w2}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V7x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V7x}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27a}})}}~{V0w1}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V2f}}~{V7x}})}})}}~{V6y}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V9y \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}})}}~{V8x}})}}~{V9y}})}})}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Estrict}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_REP}~{A\_27a}})}}~{V0w1}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{A\_27b}})}}~{({{{ap}~{V2f}}~{V8x}})}})}}~{({{{ap}~{V2f}}~{V9y}})}})}})}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Estrict}}~{A\_27b}})}}~{({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_REP}~{A\_27b}})}}~{V1w2}})}})}})}})}})}})}})}})}})}})}})}})}})}}}}}$.
\end{definition}
Let $ ty\_2Eordinal\_2Eordinal : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Eordinal\_2Eordinal}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfepgchegjgogbgmgpfcdfepgchegjgogbgmg}
\end{equation}
Let $ c\_2Eordinal\_2Eordinal\_\_ABS\_\_CLASS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eordinal\_2Eordinal\_\_ABS\_\_CLASS~\\
A\_27a \in ((ty\_2Eordinal\_2Eordinal~A\_27a)^{{({{2}^{{({{ty\_2Ewellorder\_2Ewellorder}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgchegjgogbgmgpfcdfepgchegjgogbgmgpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfepgchegjgogbgmgpfcdfepgchegjgogbgmgpfpfbecedf}
We define $\mname{c\_2Eordinal\_2Eordinal\_\_ABS}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ewellorder\_2Ewellorder}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})} .  {({{{ap}~{({{c\_2Eordinal\_2Eordinal\_\_ABS\_\_CLASS}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Ewellorder\_2Eorderiso}}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{A\_27a}})}}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}~{V0r}})}})}}}$.
\end{definition}
Let $ c\_2Eordinal\_2Eordinal\_\_REP\_\_CLASS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eordinal\_2Eordinal\_\_REP\_\_CLASS~\\
A\_27a \in ((2^{{({{ty\_2Ewellorder\_2Ewellorder}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}})^{{({{ty\_2Eordinal\_2Eordinal}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgchegjgogbgmgpfcdfepgchegjgogbgmgpfpfcffeafpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfepgchegjgogbgmgpfcdfepgchegjgogbgmgpfpfcffeaf}
We define $\mname{c\_2Eordinal\_2Eordinal\_\_REP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0a \in {({{ty\_2Eordinal\_2Eordinal}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{ty\_2Ewellorder\_2Ewellorder}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}})}}~{({{{ap}~{({{c\_2Eordinal\_2Eordinal\_\_REP\_\_CLASS}~{A\_27a}})}}~{V0a}})}})}}}$.
\end{definition}
Let $ c\_2Eordinal\_2EordADD : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Eordinal\_2EordADD~A\_27a \in ((\\
(ty\_2Eordinal\_2Eordinal~A\_27a)^{{({{ty\_2Eordinal\_2Eordinal}~{A\_27a}})}})^{{({{ty\_2Eordinal\_2Eordinal}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepgchegjgogbgmgpfcdfepgchegbeeeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfehhfgmgmgpgchegfgchpfcdfejgdhfghg}
We define $\mname{c\_2Ewellorder\_2Eiseg}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{ty\_2Ewellorder\_2Ewellorder}~{A\_27a}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2y}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2y}})}}~{V1x}})}})}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Estrict}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_REP}~{A\_27a}})}}~{V0w}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechchfgdhehchjgdgeh}
We define $\mname{c\_2Eset\_\_relation\_2Errestrict}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27a}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V3y}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V3y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3y}})}}~{V1s}})}})}})}})}})}})}})}})}}}}$.
\end{definition}
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:dgpfcdfehhfgmgmgpgchegfgchpfcdfehhpgcgpgfhogeg}
We define $\mname{c\_2Ewellorder\_2Ewobound}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1w \in {({{ty\_2Ewellorder\_2Ewellorder}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_ABS}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Errestrict}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ewellorder\_2Ewellorder\_\_REP}~{A\_27a}})}}~{V1w}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2Eiseg}}~{A\_27a}})}}~{V1w}})}}~{V0x}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchmgeh}
We define $\mname{c\_2Ewellorder\_2Eorderlt}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0w1 \in {({{ty\_2Ewellorder\_2Ewellorder}~{A\_27a}})} .  {{\uniddcgcg} V1w2 \in {({{ty\_2Ewellorder\_2Ewellorder}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V2x \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{V2x}})}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2EelsOf}}~{A\_27b}})}}~{V1w2}})}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Ewellorder\_2Eorderiso}}~{A\_27a}}~{A\_27b}})}}~{V0w1}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2Ewobound}}~{A\_27b}})}}~{V2x}})}}~{V1w2}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfepgchegjgogbgmgpfcdfepgchegmgeh}
We define $\mname{c\_2Eordinal\_2Eordlt}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0T1 \in {({{ty\_2Eordinal\_2Eordinal}~{A\_27a}})} .  {{\uniddcgcg} V1T2 \in {({{ty\_2Eordinal\_2Eordinal}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Ewellorder\_2Eorderlt}}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{A\_27a}})}}~{({{{ty\_2Esum\_2Esum}~{ty\_2Enum\_2Enum}}~{A\_27a}})}})}}~{({{{ap}~{({{\mname{c\_2Eordinal\_2Eordinal\_\_REP}}~{A\_27a}})}}~{V0T1}})}})}}~{({{{ap}~{({{\mname{c\_2Eordinal\_2Eordinal\_\_REP}}~{A\_27a}})}}~{V1T2}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfele}
We define $\mname{c\_2Ecombin\_2EK}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1y \in {A\_27b} .  {V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfedf}
We define $\mname{c\_2Ecombin\_2ES}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1g \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2x}})}}~{({{{ap}~{V1g}}~{V2x}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfeje}
We define $\mname{c\_2Ecombin\_2EI}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2ES}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}}~{A\_27a}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}})}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{A\_27a}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhpgehjgfgogehpfcdfepfcdeepfcdeepfddfe}
We define $\mname{c\_2Equotient\_2E\_2D\_2D\_3E}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} A\_27d : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27c}^{{A\_27a}}})} .  {{\uniddcgcg} V1g \in {({{A\_27d}^{{A\_27b}}})} .  {({{\uniddcgcg} V2h \in {({{A\_27b}^{{A\_27c}}})} .  {({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{V1g}}~{({{{ap}~{V2h}}~{({{{ap}~{V0f}}~{V3x}})}})}})}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhpgehjgfgogehpfcdfepfddeepfddeepfddeepfddfe}
We define $\mname{c\_2Equotient\_2E\_3D\_3D\_3D\_3E}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R1 \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1R2 \in {({{({{2}^{{A\_27b}}})}^{{A\_27b}}})} .  {{\uniddcgcg} V2f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V3g \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R1}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V1R2}}~{({{{ap}~{V2f}}~{V4x}})}})}}~{({{{ap}~{V3g}}~{V5y}})}})}})}})}})}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhpgehjgfgogehpfcdfebfffpeefjefeoeef}
We define $\mname{c\_2Equotient\_2EQUOTIENT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1abs \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V2rep \in {({{A\_27a}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V3a \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V1abs}}~{({{{ap}~{V2rep}}~{V3a}})}})}})}}~{V3a}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V4a \in {A\_27b} .  {({{{ap}~{({{{ap}~{V0R}}~{({{{ap}~{V2rep}}~{V4a}})}})}}~{({{{ap}~{V2rep}}~{V4a}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5r \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6s \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V5r}})}}~{V6s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V5r}})}}~{V5r}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V6s}})}}~{V6s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V1abs}}~{V5r}})}})}}~{({{{ap}~{V1abs}}~{V6s}})}})}})}})}})}})}})}})}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfehf}
We define $\mname{c\_2Ecombin\_2EW}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27b}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V1x}})}}~{V1x}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhpgehjgfgogehpfcdfechfgdhahfgdgehdh}
We define $\mname{c\_2Equotient\_2Erespects}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{{\mname{c\_2Ecombin\_2EW}}~{A\_27a}}~{A\_27b}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfecffedfpfpfgepecfbememe}
We define $\mname{c\_2Ebool\_2ERES\_\_FORALL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0p \in {({{2}^{{A\_27a}}})} .  {({{\uniddcgcg} V1m \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0p}})}})}}~{({{{ap}~{V1m}}~{V2x}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhpgehjgfgogehpfcdfefebfffjegf}
We define $\mname{c\_2Equotient\_2EEQUIV}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0E \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{{ap}~{V0E}}~{V1x}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{({{{ap}~{V0E}}~{V1x}})}})}}~{({{{ap}~{V0E}}~{V2y}})}})}})}})}})}})}})}}}$.
\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{\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{\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}
\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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1g \in (A\_27b^{{A\_27a}}) .  ((V0f=\\
V1g){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  ((ap~V0f~V2x)=(ap~V1g~V2x))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegeffoepfpffebfpfpfefiene}
\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} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjded}(p~V1t2)){\unicdbdjded}(((p~\\
V0t1){\unicdbdjdcd}(p~V1t2)){\unicdcdcdhd}((p~V1t2){\unicdbdjdcd}(p~V0t1))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfjeneafpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1x\_27 \in 2 .  ({\unicdcdadad} V2y \in 2 .  ({\unicdcdadad} V3y\_27 \in \\
2 .  ((((p~V0x){\unicdbdjded}(p~V1x\_27)){\unicdcdcdhd}((p~V1x\_27){\unicdbdjdcd}((p~V2y){\unicdbdjded}(p~V3y\_27)))){\unicdbdjdcd}\\
(((p~V0x){\unicdbdjdcd}(p~V2y)){\unicdbdjded}((p~V1x\_27){\unicdbdjdcd}(p~V3y\_27))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((ap~(\mname{c\_2Ecombin\_2EI}~\\
A\_27a)~V0x)=V0x))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfejepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
(ty\_2Ewellorder\_2Ewellorder~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~\\
A\_27a))~(ty\_2Eordinal\_2Eordinal~A\_27a))~(\mname{c\_2Ewellorder\_2Eorderiso}~\\
(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~A\_27a)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~\\
A\_27a)))~(\mname{c\_2Eordinal\_2Eordinal\_\_ABS}~A\_27a))~(\mname{c\_2Eordinal\_2Eordinal\_\_REP}~\\
A\_27a)))\end{array}
\label{hyp:ehigngpfcdfepgchegjgogbgmgpfcdfepgchegjgogbgmgpfpfbfffpeefjefeoeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0b \in (ty\_2Eordinal\_2Eordinal~\\
A\_27a) .  ({\unicdcdadad} V1a \in (ty\_2Eordinal\_2Eordinal~A\_27a) .  ({\unicdcdadad} V2c \in \\
(ty\_2Eordinal\_2Eordinal~A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Eordinal\_2Eordlt}~\\
A\_27a)~V1a)~V0b)){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Eordinal\_2Eordlt}~A\_27a)~(ap~(ap~\\
(c\_2Eordinal\_2EordADD~A\_27a)~V2c)~V1a))~(ap~(ap~(c\_2Eordinal\_2EordADD~\\
A\_27a)~V2c)~V0b)))))))\end{array}
\label{hyp:ehigngpfcdfepgchegjgogbgmgpfcdfepgchegbeeeeepfpfnepeoepe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27a~A\_27a)~(\mname{c\_2Emin\_2E\_3D}~A\_27a))~(\mname{c\_2Ecombin\_2EI}~A\_27a))~(\\
\mname{c\_2Ecombin\_2EI}~A\_27a)))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfejeeefeoeefjeefjfpfpfbfffpeefjefeoeef}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0R1 \in (\\
(2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs1 \in (A\_27c^{{A\_27a}}) .  ({\unicdcdadad} V2rep1 \in \\
(A\_27a^{{A\_27c}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~A\_27a~A\_27c)~\\
V0R1)~V1abs1)~V2rep1)){\unicdbdjdcd}({\unicdcdadad} V3R2 \in ((2^{{A\_27b}})^{{A\_27b}}) .  ({\unicdcdadad} V4abs2 \in \\
(A\_27d^{{A\_27b}}) .  ({\unicdcdadad} V5rep2 \in (A\_27b^{{A\_27d}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27b~A\_27d)~V3R2)~V4abs2)~V5rep2)){\unicdbdjdcd}(p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
(A\_27b^{{A\_27a}})~(A\_27d^{{A\_27c}}))~(ap~(ap~(\mname{c\_2Equotient\_2E\_3D\_3D\_3D\_3E}~\\
A\_27a~A\_27b)~V0R1)~V3R2))~(ap~(ap~(\mname{c\_2Equotient\_2E\_2D\_2D\_3E}~A\_27c~\\
A\_27b~A\_27a~A\_27d)~V2rep1)~V4abs2))~(ap~(ap~(\mname{c\_2Equotient\_2E\_2D\_2D\_3E}~\\
A\_27a~A\_27d~A\_27c~A\_27b)~V1abs1)~V5rep2)))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfegeffoepfpfbfffpeefjefeoeef}
\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} V0R \in ((2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V2rep \in (A\_27a^{{A\_27b}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27a~A\_27b)~V0R)~V1abs)~V2rep)){\unicdbdjdcd}({\unicdcdadad} V3x \in A\_27b .  ({\unicdcdadad} V4y \in \\
A\_27b .  ((V3x=V4y){\unicdbdjded}(p~(ap~(ap~V0R~(ap~V2rep~V3x))~(ap~V2rep~V4y))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfefebfffbemedfpfpfafcfdf}
\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} V0R \in ((2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V2rep \in (A\_27a^{{A\_27b}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27a~A\_27b)~V0R)~V1abs)~V2rep)){\unicdbdjdcd}({\unicdcdadad} V3x1 \in A\_27a .  ({\unicdcdadad} V4x2 \in \\
A\_27a .  ({\unicdcdadad} V5y1 \in A\_27a .  ({\unicdcdadad} V6y2 \in A\_27a .  (((p~(ap~(ap~V0R~\\
V3x1)~V4x2)){\unicdcdcdhd}(p~(ap~(ap~V0R~V5y1)~V6y2))){\unicdbdjdcd}((p~(ap~(ap~V0R~V3x1)~V5y1)){\unicdbdjded}\\
(p~(ap~(ap~V0R~V4x2)~V6y2))))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfefebfffbemedfpfpfcfdfaf}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0R1 \in (\\
(2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs1 \in (A\_27c^{{A\_27a}}) .  ({\unicdcdadad} V2rep1 \in \\
(A\_27a^{{A\_27c}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~A\_27a~A\_27c)~\\
V0R1)~V1abs1)~V2rep1)){\unicdbdjdcd}({\unicdcdadad} V3R2 \in ((2^{{A\_27b}})^{{A\_27b}}) .  ({\unicdcdadad} V4abs2 \in \\
(A\_27d^{{A\_27b}}) .  ({\unicdcdadad} V5rep2 \in (A\_27b^{{A\_27d}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27b~A\_27d)~V3R2)~V4abs2)~V5rep2)){\unicdbdjdcd}({\unicdcdadad} V6f \in (A\_27d^{{A\_27c}}) .  \\
(({\uniddcgcg} V7x \in A\_27c .  (ap~V6f~V7x))=(ap~(ap~(ap~(\mname{c\_2Equotient\_2E\_2D\_2D\_3E}~\\
A\_27c~A\_27b~A\_27a~A\_27d)~V2rep1)~V4abs2)~({\uniddcgcg} V8x \in A\_27a .  (ap~V5rep2~\\
(ap~V6f~(ap~V1abs1~V8x)))))))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfemebeneceeebepfpfafcfdf}
\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} V0REL \in ((2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V2rep \in (A\_27a^{{A\_27b}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27a~A\_27b)~V0REL)~V1abs)~V2rep)){\unicdbdjdcd}({\unicdcdadad} V3x1 \in A\_27a .  ({\unicdcdadad} V4x2 \in \\
A\_27a .  ((p~(ap~(ap~V0REL~V3x1)~V4x2)){\unicdbdjdcd}(p~(ap~(ap~V0REL~V3x1)~(ap~V2rep~\\
(ap~V1abs~V4x2)))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfecffeafpfpfbecedfpfpfcfdfaf}
\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} V0R \in ((2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V2rep \in (A\_27a^{{A\_27b}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27a~A\_27b)~V0R)~V1abs)~V2rep)){\unicdbdjdcd}({\unicdcdadad} V3f \in (2^{{A\_27b}}) .  ((p~(\\
ap~(\mname{c\_2Ebool\_2E\_21}~A\_27b)~V3f)){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebool\_2ERES\_\_FORALL}~\\
A\_27a)~(ap~(\mname{c\_2Equotient\_2Erespects}~A\_27a~2)~V0R))~(ap~(ap~(ap~\\
(\mname{c\_2Equotient\_2E\_2D\_2D\_3E}~A\_27a~2~A\_27b~2)~V1abs)~(\mname{c\_2Ecombin\_2EI}~\\
2))~V3f)))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfegepecfbememepfpfafcfdf}
\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} V0R \in ((2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V2rep \in (A\_27a^{{A\_27b}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27a~A\_27b)~V0R)~V1abs)~V2rep)){\unicdbdjdcd}({\unicdcdadad} V3f \in (2^{{A\_27a}}) .  ({\unicdcdadad} V4g \in \\
(2^{{A\_27a}}) .  ((p~(ap~(ap~(ap~(ap~(\mname{c\_2Equotient\_2E\_3D\_3D\_3D\_3E}~A\_27a~\\
2)~V0R)~(\mname{c\_2Emin\_2E\_3D}~2))~V3f)~V4g)){\unicdbdjdcd}((p~(ap~(ap~(\mname{c\_2Ebool\_2ERES\_\_FORALL}~\\
A\_27a)~(ap~(\mname{c\_2Equotient\_2Erespects}~A\_27a~2)~V0R))~V3f)){\unicdbdjded}(p~(\\
ap~(ap~(\mname{c\_2Ebool\_2ERES\_\_FORALL}~A\_27a)~(ap~(\mname{c\_2Equotient\_2Erespects}~\\
A\_27a~2)~V0R))~V4g))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfecffedfpfpfgepecfbememepfpfcfdfaf}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0R1 \in (\\
(2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V1abs1 \in (A\_27c^{{A\_27a}}) .  ({\unicdcdadad} V2rep1 \in \\
(A\_27a^{{A\_27c}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~A\_27a~A\_27c)~\\
V0R1)~V1abs1)~V2rep1)){\unicdbdjdcd}({\unicdcdadad} V3R2 \in ((2^{{A\_27b}})^{{A\_27b}}) .  ({\unicdcdadad} V4abs2 \in \\
(A\_27d^{{A\_27b}}) .  ({\unicdcdadad} V5rep2 \in (A\_27b^{{A\_27d}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Equotient\_2EQUOTIENT}~\\
A\_27b~A\_27d)~V3R2)~V4abs2)~V5rep2)){\unicdbdjdcd}({\unicdcdadad} V6f \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V7g \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V8x \in A\_27a .  ({\unicdcdadad} V9y \in \\
A\_27a .  (((p~(ap~(ap~(ap~(ap~(\mname{c\_2Equotient\_2E\_3D\_3D\_3D\_3E}~A\_27a~\\
A\_27b)~V0R1)~V3R2)~V6f)~V7g)){\unicdcdcdhd}(p~(ap~(ap~V0R1~V8x)~V9y))){\unicdbdjdcd}(p~(ap~(\\
ap~V3R2~(ap~V6f~V8x))~(ap~V7g~V9y))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfebeafafmejfpfpfcfdfaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0E \in ((2^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V1P \in (2^{{A\_27a}}) .  ((p~(ap~(\mname{c\_2Equotient\_2EEQUIV}~A\_27a)~\\
V0E)){\unicdbdjdcd}((p~(ap~(ap~(\mname{c\_2Ebool\_2ERES\_\_FORALL}~A\_27a)~(ap~(\mname{c\_2Equotient\_2Erespects}~\\
A\_27a~2)~V0E))~V1P)){\unicdbdjded}(p~(ap~(\mname{c\_2Ebool\_2E\_21}~A\_27a)~V1P))))))\end{array}
\label{hyp:ehigngpfcdfebhfhpgehjgfgogehpfcdfefebfffjegfpfpfcffedfpfpfgepecfbememe}
\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} V0w \in (ty\_2Ewellorder\_2Ewellorder~\\
A\_27a) .  (p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderiso}~A\_27a~A\_27a)~V0w)~\\
V0w)))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchjgdhpgpfpfcffegeme}
\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} V0w1 \in (ty\_2Ewellorder\_2Ewellorder~A\_27a) .  ({\unicdcdadad} V1w2 \in \\
(ty\_2Ewellorder\_2Ewellorder~A\_27b) .  ((p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderiso}~\\
A\_27a~A\_27b)~V0w1)~V1w2)){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderiso}~\\
A\_27b~A\_27a)~V1w2)~V0w1)))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchjgdhpgpfpfdfjfne}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0w1 \in (ty\_2Ewellorder\_2Ewellorder~A\_27a) .  \\
({\unicdcdadad} V1w2 \in (ty\_2Ewellorder\_2Ewellorder~A\_27b) .  ({\unicdcdadad} V2w3 \in \\
(ty\_2Ewellorder\_2Ewellorder~A\_27c) .  (((p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderiso}~\\
A\_27a~A\_27b)~V0w1)~V1w2)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderiso}~\\
A\_27b~A\_27c)~V1w2)~V2w3))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderiso}~\\
A\_27a~A\_27c)~V0w1)~V2w3))))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchjgdhpgpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0w \in (ty\_2Ewellorder\_2Ewellorder~\\
A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderlt}~A\_27a~A\_27a)~V0w)~\\
V0w)){\unicdbdjded}False))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchmgehpfpfcffegeme}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0w1 \in (ty\_2Ewellorder\_2Ewellorder~A\_27a) .  \\
({\unicdcdadad} V1w2 \in (ty\_2Ewellorder\_2Ewellorder~A\_27b) .  ({\unicdcdadad} V2w3 \in \\
(ty\_2Ewellorder\_2Ewellorder~A\_27c) .  (((p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderlt}~\\
A\_27a~A\_27b)~V0w1)~V1w2)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderlt}~\\
A\_27b~A\_27c)~V1w2)~V2w3))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderlt}~\\
A\_27a~A\_27c)~V0w1)~V2w3))))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchmgehpfpfefcfbeoedf}
\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} V0w1 \in (ty\_2Ewellorder\_2Ewellorder~A\_27a) .  ({\unicdcdadad} V1w2 \in \\
(ty\_2Ewellorder\_2Ewellorder~A\_27b) .  ((p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderlt}~\\
A\_27a~A\_27b)~V0w1)~V1w2)){\unicdcdcdid}((p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderiso}~\\
A\_27a~A\_27b)~V0w1)~V1w2)){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ewellorder\_2Eorderlt}~\\
A\_27b~A\_27a)~V1w2)~V0w1))))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchmgehpfpfehchjgdgigpgehpgngjh}
\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} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0x0 \in (\\
ty\_2Ewellorder\_2Ewellorder~A\_27a) .  ({\unicdcdadad} V1y0 \in (ty\_2Ewellorder\_2Ewellorder~\\
A\_27b) .  ({\unicdcdadad} V2a0 \in (ty\_2Ewellorder\_2Ewellorder~A\_27c) .  (\\
{\unicdcdadad} V3b0 \in (ty\_2Ewellorder\_2Ewellorder~A\_27d) .  (((p~(ap~(ap~\\
(\mname{c\_2Ewellorder\_2Eorderiso}~A\_27a~A\_27b)~V0x0)~V1y0)){\unicdcdcdhd}(p~(ap~(ap~\\
(\mname{c\_2Ewellorder\_2Eorderiso}~A\_27c~A\_27d)~V2a0)~V3b0))){\unicdbdjdcd}((p~(ap~\\
(ap~(\mname{c\_2Ewellorder\_2Eorderlt}~A\_27a~A\_27c)~V0x0)~V2a0)){\unicdbdjded}(p~(ap~\\
(ap~(\mname{c\_2Ewellorder\_2Eorderlt}~A\_27b~A\_27d)~V1y0)~V3b0))))))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfepgchegfgchmgehpfpfpgchegfgchjgdhpg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfepgchegjgogbgmgpfcdfepgchegmgehpfpfdebeoedefeme}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0b \in (ty\_2Eordinal\_2Eordinal~\\
A\_27a) .  ({\unicdcdadad} V1a \in (ty\_2Eordinal\_2Eordinal~A\_27a) .  ({\unicdcdadad} V2c \in \\
(ty\_2Eordinal\_2Eordinal~A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Eordinal\_2Eordlt}~\\
A\_27a)~(ap~(ap~(c\_2Eordinal\_2EordADD~A\_27a)~V2c)~V1a))~(ap~(ap~(\\
c\_2Eordinal\_2EordADD~A\_27a)~V2c)~V0b))){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Eordinal\_2Eordlt}~\\
A\_27a)~V1a)~V0b))))))\end{array}$$
\end{theorem}
\end{document}
