\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\_2Eset\_\_relation\_2Ereln\_\_rel\_\_conv\_\_thms (TMWXt5mY4FEMojkyhdtSLepu5hKMtLUB8ym)}
\maketitle
\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:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def: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\_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}
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}
\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:dgpfcdfechfgmgbgehjgpgogpfcdfecfeepene}
We define $\mname{c\_2Erelation\_2ERDOM}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V2y \in {A\_27b} .  {({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V2y}})}})}})}}}}}$.
\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: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:dgpfcdfechfgmgbgehjgpgogpfcdfecfcfbeoehefe}
We define $\mname{c\_2Erelation\_2ERRANGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0R}}~{V2x}})}}~{V1y}})}})}})}}}}}$.
\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: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: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:dgpfcdfechfgmgbgehjgpgogpfcdfedfefcfpecfee}
We define $\mname{c\_2Erelation\_2ESTRORD}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V1a}})}}~{V2b}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1a}})}}~{V2b}})}})}})}}}}}$.
\end{definition}
\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: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}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfepe}
We define $\mname{c\_2Erelation\_2EO}$ to be ${{\uniddcgcg} A\_27g : {\iota} .  {{\uniddcgcg} A\_27h : {\iota} .  {{\uniddcgcg} A\_27k : {\iota} .  {{\uniddcgcg} V0R1 \in {({{({{2}^{{A\_27k}}})}^{{A\_27h}}})} .  {{\uniddcgcg} V1R2 \in {({{({{2}^{{A\_27h}}})}^{{A\_27g}}})} .  {{\uniddcgcg} V2x \in {A\_27g} .  {{\uniddcgcg} V3z \in {A\_27k} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27h}})}}~{({{\uniddcgcg} V4y \in {A\_27h} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V1R2}}~{V2x}})}}~{V4y}})}})}}~{({{{ap}~{({{{ap}~{V0R1}}~{V4y}})}}~{V3z}})}})}})}})}}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechdgpgngah}
We define $\mname{c\_2Eset\_\_relation\_2Ercomp}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0r1 \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27c}})}}})} .  {{\uniddcgcg} V1r2 \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27b}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27b}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}~{2}})}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27b}})}}~{V2x}})}}~{V3y}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27c}})}}~{({{\uniddcgcg} V4z \in {A\_27c} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27c}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27c}})}}~{V2x}})}}~{V4z}})}})}}~{V0r1}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27c}}~{A\_27b}})}}~{V4z}})}}~{V3y}})}})}}~{V1r2}})}})}})}})}})}})}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfecfcfffoejegf}
We define $\mname{c\_2Eset\_\_relation\_2ERRUNIV}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27a} .  {({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V1x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2y}})}}~{V0s}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfefhogjgghpfpfchfgmgog}
We define $\mname{c\_2Eset\_\_relation\_2Euniv\_\_reln}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0xs \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27a}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}})}}~{({{\uniddcgcg} V1x1 \in {A\_27a} .  {({{\uniddcgcg} V2x2 \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x1}})}}~{V2x2}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V1x1}})}}~{V0xs}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x2}})}}~{V0xs}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgmgpfpfehpgpfpfchfgmgog}
We define $\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27b}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}~{2}})}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{\uniddcgcg} V2y \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27b}})}}~{V1x}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V2y}})}})}})}})}})}})}}}}$.
\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:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeehdg}
We define $\mname{c\_2Eset\_\_relation\_2Etc}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{\uniddcgcg} V1a0 \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})}})}}~{({{\uniddcgcg} V2tc\_27 \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{\uniddcgcg} V3a0 \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{V3a0}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V4x}})}}~{V5y}})}})}})}}~{({{{ap}~{V0r}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V6x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V7y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{V3a0}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V6x}})}}~{V7y}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V8z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2tc\_27}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V6x}})}}~{V8z}})}})}})}}~{({{{ap}~{V2tc\_27}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V8z}})}}~{V7y}})}})}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{V2tc\_27}}~{V3a0}})}})}})}})}})}}~{({{{ap}~{V2tc\_27}}~{V1a0}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeefde}
We define $\mname{c\_2Erelation\_2ETC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3P \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V7y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V7y}})}}~{V8z}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V8z}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V1a}})}}~{V2b}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfebgdgjhdgmgjgdg}
We define $\mname{c\_2Eset\_\_relation\_2Eacyclic}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V1x}})}})}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Etc}}~{A\_27a}})}}~{V0r}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfecffemepfpfcffedfefcfjedeef}
We define $\mname{c\_2Epred\_\_set\_2EREL\_\_RESTRICT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3y}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V2x}})}}~{V3y}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfejgchchfgggmgfgihjgghfg}
We define $\mname{c\_2Erelation\_2Eirreflexive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V1x}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfejgchchfgggmgfgihjgghfg}
We define $\mname{c\_2Eset\_\_relation\_2Eirreflexive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V2x}})}})}}~{V0r}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfecfffoejepeoe}
We define $\mname{c\_2Erelation\_2ERUNION}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R1 \in {({{({{2}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1R2 \in {({{({{2}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{V0R1}}~{V2x}})}}~{V3y}})}})}}~{({{{ap}~{({{{ap}~{V1R2}}~{V2x}})}}~{V3y}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfecfcffegemepfpffeifaf}
We define $\mname{c\_2Eset\_\_relation\_2ERREFL\_\_EXP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERUNION}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}}~{V3y}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfechfgggmgfgihjgghfg}
We define $\mname{c\_2Erelation\_2Ereflexive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V1x}})}})}})}}}$.
\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:dgpfcdfechfgmgbgehjgpgogpfcdfeehchbgogdhjgehjgghfg}
We define $\mname{c\_2Erelation\_2Etransitive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V2y}})}}~{V3z}})}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V3z}})}})}})}})}})}})}})}})}}}$.
\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:dgpfcdfechfgmgbgehjgpgogpfcdfebgogehjgdhjhngngfgehchjgdg}
We define $\mname{c\_2Erelation\_2Eantisymmetric}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V0R}}~{V1x}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{V0R}}~{V2y}})}}~{V1x}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}}}$.
\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:dgpfcdfechfgmgbgehjgpgogpfcdfehffgbglgpechegfgch}
We define $\mname{c\_2Erelation\_2EWeakOrder}$ to be ${{\uniddcgcg} A\_27g : {\iota} .  {{\uniddcgcg} V0Z \in {({{({{2}^{{A\_27g}}})}^{{A\_27g}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2Ereflexive}}~{A\_27g}})}}~{V0Z}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2Eantisymmetric}}~{A\_27g}})}}~{V0Z}})}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2Etransitive}}~{A\_27g}})}}~{V0Z}})}})}})}}}$.
\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:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeahbgchehjgbgmgpfpfpgchegfgch}
We define $\mname{c\_2Eset\_\_relation\_2Epartial\_\_order}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0r}})}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0r}})}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Etransitive}}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Ereflexive}}~{A\_27a}})}}~{V0r}})}}~{V1s}})}})}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Eantisym}}~{A\_27a}})}}~{V0r}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeehchjgdgigpgehpgngpgfhdh}
We define $\mname{c\_2Erelation\_2Etrichotomous}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1a \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{V0R}}~{V1a}})}}~{V2b}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{V0R}}~{V2b}})}}~{V1a}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1a}})}}~{V2b}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehffgbglgmejgogfgbgchpechegfgch}
We define $\mname{c\_2Erelation\_2EWeakLinearOrder}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2EWeakOrder}}~{A\_27a}})}}~{V0R}})}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2Etrichotomous}}~{A\_27a}})}}~{V0R}})}})}}}$.
\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:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgmgogpfpfehpgpfpfchfgmg}
We define $\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}})} .  {({{\uniddcgcg} V1x \in {A\_27a} .  {({{\uniddcgcg} V2y \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27b}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfedfehchpgoghgpechegfgch}
We define $\mname{c\_2Erelation\_2EStrongOrder}$ to be ${{\uniddcgcg} A\_27g : {\iota} .  {{\uniddcgcg} V0Z \in {({{({{2}^{{A\_27g}}})}^{{A\_27g}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2Eirreflexive}}~{A\_27g}})}}~{V0Z}})}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2Etransitive}}~{A\_27g}})}}~{V0Z}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfedfehchpgoghgmejgogfgbgchpechegfgch}
We define $\mname{c\_2Erelation\_2EStrongLinearOrder}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2EStrongOrder}}~{A\_27a}})}}~{V0R}})}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2Etrichotomous}}~{A\_27a}})}}~{V0R}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejegf}
We define $\mname{c\_2Epred\_\_set\_2EUNIV}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfedhehchjgdgehpfpfmgjgogfgbgchpfpfpgchegfgch}
We define $\mname{c\_2Eset\_\_relation\_2Estrict\_\_linear\_\_order}$ 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\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{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}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4y \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}})}}~{V3x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V4y}})}}~{V1s}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V4y}})}})}})}})}})}}~{({{{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}})}}~{V3x}})}}~{V4y}})}})}}~{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}})}}~{V4y}})}}~{V3x}})}})}}~{V0r}})}})}})}})}})}})}})}})}})}})}})}}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0R \in ((2^{{A\_27a}})^{{A\_27a}}) .  \\
((ap~(ap~(\mname{c\_2Eset\_\_relation\_2ERREFL\_\_EXP}~A\_27a)~V0R)~(\mname{c\_2Epred\_\_set\_2EUNIV}~\\
A\_27a))=V0R))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfecfcffegemepfpffeifafpfpfffoejegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0R \in ((2^{{A\_27a}})^{{A\_27a}}) .  \\
((ap~(ap~(\mname{c\_2Epred\_\_set\_2EREL\_\_RESTRICT}~A\_27a)~V0R)~(\mname{c\_2Epred\_\_set\_2EUNIV}~\\
A\_27a))=V0R))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfecffemepfpfcffedfefcfjedeefpfpfffoejegf}
\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} V0xy \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ({\unicdcdadad} V1R \in ((\\
2^{{A\_27b}})^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b))~V0xy)~(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~\\
A\_27b)~V1R))){\unicdbdjded}(p~(ap~(ap~V1R~(ap~(c\_2Epair\_2EFST~A\_27a~A\_27b)~V0xy))~\\
(ap~(c\_2Epair\_2ESND~A\_27a~A\_27b)~V0xy))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfejgogpfpfchfgmgpfpfehpgpfpfchfgmgog}
\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^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  ({\unicdcdadad} V1x \in \\
A\_27a .  ({\unicdcdadad} V2y \in A\_27b .  ((p~(ap~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27b)~V0r)~V1x)~V2y)){\unicdbdjded}(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)~V1x)~V2y))~V0r))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgmgogpfpfehpgpfpfchfgmgpfpfbgahah}
\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\_27b}})^{{A\_27a}}) .  ((ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~A\_27b)~\\
V0R))=V0R))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgmgpfpfehpgpfpfchfgmgogpfpfjgoggh}
\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^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  ((ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27b)~\\
V0r))=V0r))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgmgogpfpfehpgpfpfchfgmgpfpfjgoggh}
\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} V0r1 \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  ({\unicdcdadad} V1r2 \in \\
(2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  (((ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27b)~V0r1)=(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~\\
A\_27b)~V1r2)){\unicdbdjded}(V0r1=V1r2))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgmgogpfpfehpgpfpfchfgmgpfpfbdbd}
\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} V0R1 \in ((2^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1R2 \in ((2^{{A\_27b}})^{{A\_27a}}) .  \\
(((ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~A\_27b)~V0R1)=\\
(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~A\_27b)~V1R2)){\unicdbdjded}\\
(V0R1=V1R2))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgmgpfpfehpgpfpfchfgmgogpfpfbdbd}
\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^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  ((ap~(\mname{c\_2Eset\_\_relation\_2Edomain}~\\
A\_27a~A\_27b)~V0r)=(ap~(\mname{c\_2Erelation\_2ERDOM}~A\_27a~A\_27b)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27b)~V0r))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeegpgngbgjgogpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\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^{{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}}) .  ((ap~(\mname{c\_2Eset\_\_relation\_2Erange}~\\
A\_27a~A\_27b)~V0r)=(ap~(\mname{c\_2Erelation\_2ERRANGE}~A\_27b~A\_27a)~(ap~\\
(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27b~A\_27a)~V0r))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechbgoghgfgpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((ap~(\mname{c\_2Eset\_\_relation\_2Estrict}~A\_27a)~V0r)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~\\
A\_27a~A\_27a)~(ap~(\mname{c\_2Erelation\_2ESTRORD}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V0r)))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfedhehchjgdgehpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ((ap~(ap~(\mname{c\_2Eset\_\_relation\_2Errestrict}~\\
A\_27a)~V0r)~V1s)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EREL\_\_RESTRICT}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V0r))~V1s)))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechchfgdhehchjgdgehpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\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} V0r1 \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27c}})}}) .  \\
({\unicdcdadad} V1r2 \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27b}})}}) .  ((ap~(ap~(\mname{c\_2Eset\_\_relation\_2Ercomp}~\\
A\_27a~A\_27b~A\_27c)~V0r1)~V1r2)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~\\
A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Erelation\_2EO}~A\_27a~A\_27c~A\_27b)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27c~A\_27b)~V1r2))~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~\\
A\_27c)~V0r1))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechdgpgngahpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ((ap~(\mname{c\_2Eset\_\_relation\_2Euniv\_\_reln}~\\
A\_27a)~V0s)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~A\_27a)~\\
(ap~(\mname{c\_2Eset\_\_relation\_2ERRUNIV}~A\_27a)~V0s))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfefhogjgghpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((ap~(\mname{c\_2Eset\_\_relation\_2Etc}~A\_27a)~V0r)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~\\
A\_27a~A\_27a)~(ap~(\mname{c\_2Erelation\_2ETC}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V0r)))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeehdgpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((p~(ap~(\mname{c\_2Eset\_\_relation\_2Eacyclic}~A\_27a)~V0r)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Eirreflexive}~\\
A\_27a)~(ap~(\mname{c\_2Erelation\_2ETC}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V0r))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfebgdgjhdgmgjgdgpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Eirreflexive}~\\
A\_27a)~V0r)~V1s)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Eirreflexive}~A\_27a)~(\\
ap~(ap~(\mname{c\_2Epred\_\_set\_2EREL\_\_RESTRICT}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V0r))~V1s))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfejgchchfgggmgfgihjgghfgpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Ereflexive}~\\
A\_27a)~V0r)~V1s)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Ereflexive}~A\_27a)~(ap~\\
(ap~(\mname{c\_2Eset\_\_relation\_2ERREFL\_\_EXP}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V0r))~V1s))))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgggmgfgihjgghfgpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((p~(ap~(\mname{c\_2Eset\_\_relation\_2Etransitive}~A\_27a)~V0r)){\unicdbdjded}(p~(ap~(\\
\mname{c\_2Erelation\_2Etransitive}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V0r)))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeehchbgogdhjgehjgghfgpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((p~(ap~(\mname{c\_2Eset\_\_relation\_2Eantisym}~A\_27a)~V0r)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Eantisymmetric}~\\
A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27a)~V0r)))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfebgogehjgdhjhngpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgoggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Epartial\_\_order}~A\_27a)~V0r)~(\\
\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2EWeakOrder}~\\
A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27a)~V0r)))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeahbgchehjgbgmgpfpfpgchegfgchpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgogghpfpfffoejegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~A\_27a)~V0r)~(\mname{c\_2Epred\_\_set\_2EUNIV}~\\
A\_27a))){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2EWeakLinearOrder}~A\_27a)~(ap~(\\
\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27a)~V0r)))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfemgjgogfgbgchpfpfpgchegfgchpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgogghpfpfffoejegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Estrict\_\_linear\_\_order}~A\_27a)~\\
V0r)~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2EStrongLinearOrder}~\\
A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27a)~V0r)))))\end{array}
\label{hyp:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfedhehchjgdgehpfpfmgjgogfgbgchpfpfpgchegfgchpfpfchfgmgogpfpfehpgpfpfchfgmgpfpfdgpgogghpfpfffoejegf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgmgogpfpfchfgmgpfpfdgpgogghpfpfehigngdh}
$$\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} V0xy \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  \\
({\unicdcdadad} V1R \in ((2^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V2r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  \\
({\unicdcdadad} V3x \in A\_27a .  ({\unicdcdadad} V4y \in A\_27b .  ({\unicdcdadad} V5r1 \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  \\
({\unicdcdadad} V6r2 \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  ({\unicdcdadad} V7R1 \in \\
((2^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V8R2 \in ((2^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V9R \in \\
((2^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V10r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27b}}~{A\_27a}})}}) .  \\
({\unicdcdadad} V11r \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  ({\unicdcdadad} V12s \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V13r1 \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27c}})}}) .  \\
({\unicdcdadad} V14r2 \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27c}}~{A\_27b}})}}) .  ((((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~V0xy)~(ap~(\\
\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~A\_27b)~V1R))){\unicdbdjded}(p~(ap~\\
(ap~V1R~(ap~(c\_2Epair\_2EFST~A\_27a~A\_27b)~V0xy))~(ap~(c\_2Epair\_2ESND~\\
A\_27a~A\_27b)~V0xy)))){\unicdcdcdhd}(((p~(ap~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27b)~V2r)~V3x)~V4y)){\unicdbdjded}(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)~V3x)~V4y))~V2r))){\unicdcdcdhd}\\
(((ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27b)~(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~\\
A\_27a~A\_27b)~V1R))=V1R){\unicdcdcdhd}(((ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27b)~\\
V2r))=V2r){\unicdcdcdhd}((((ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~\\
A\_27b)~V5r1)=(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27b)~\\
V6r2)){\unicdbdjded}(V5r1=V6r2)){\unicdcdcdhd}(((ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~\\
A\_27a~A\_27b)~V7R1)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~\\
A\_27b)~V8R2)){\unicdbdjded}(V7R1=V8R2))))))){\unicdcdcdhd}(((ap~(ap~(\mname{c\_2Eset\_\_relation\_2ERREFL\_\_EXP}~\\
A\_27a)~V9R)~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))=V9R){\unicdcdcdhd}(((ap~(ap~(\mname{c\_2Epred\_\_set\_2EREL\_\_RESTRICT}~\\
A\_27a)~V9R)~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))=V9R){\unicdcdcdhd}(((ap~(\mname{c\_2Eset\_\_relation\_2Edomain}~\\
A\_27a~A\_27b)~V2r)=(ap~(\mname{c\_2Erelation\_2ERDOM}~A\_27a~A\_27b)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27b)~V2r))){\unicdcdcdhd}(((ap~(\mname{c\_2Eset\_\_relation\_2Erange}~A\_27a~A\_27b)~\\
V10r)=(ap~(\mname{c\_2Erelation\_2ERRANGE}~A\_27b~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27b~A\_27a)~V10r))){\unicdcdcdhd}(((ap~(\mname{c\_2Eset\_\_relation\_2Estrict}~A\_27a)~\\
V11r)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~A\_27a)~(\\
ap~(\mname{c\_2Erelation\_2ESTRORD}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V11r)))){\unicdcdcdhd}(((ap~(ap~(\mname{c\_2Eset\_\_relation\_2Errestrict}~\\
A\_27a)~V11r)~V12s)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EREL\_\_RESTRICT}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V11r))~V12s))){\unicdcdcdhd}(((ap~(ap~(\mname{c\_2Eset\_\_relation\_2Ercomp}~\\
A\_27a~A\_27b~A\_27c)~V13r1)~V14r2)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~\\
A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Erelation\_2EO}~A\_27a~A\_27c~A\_27b)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27c~A\_27b)~V14r2))~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27c)~V13r1)))){\unicdcdcdhd}(((ap~(\mname{c\_2Eset\_\_relation\_2Euniv\_\_reln}~\\
A\_27a)~V12s)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~A\_27a)~\\
(ap~(\mname{c\_2Eset\_\_relation\_2ERRUNIV}~A\_27a)~V12s))){\unicdcdcdhd}(((ap~(\mname{c\_2Eset\_\_relation\_2Etc}~\\
A\_27a)~V11r)=(ap~(\mname{c\_2Eset\_\_relation\_2Erel\_\_to\_\_reln}~A\_27a~A\_27a)~\\
(ap~(\mname{c\_2Erelation\_2ETC}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V11r)))){\unicdcdcdhd}(((p~(ap~(\mname{c\_2Eset\_\_relation\_2Eacyclic}~\\
A\_27a)~V11r)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Eirreflexive}~A\_27a)~(ap~(\\
\mname{c\_2Erelation\_2ETC}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V11r))))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Eirreflexive}~\\
A\_27a)~V11r)~V12s)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Eirreflexive}~A\_27a)~\\
(ap~(ap~(\mname{c\_2Epred\_\_set\_2EREL\_\_RESTRICT}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V11r))~V12s)))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Ereflexive}~\\
A\_27a)~V11r)~V12s)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Ereflexive}~A\_27a)~(\\
ap~(ap~(\mname{c\_2Eset\_\_relation\_2ERREFL\_\_EXP}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V11r))~V12s)))){\unicdcdcdhd}(((p~(ap~(\mname{c\_2Eset\_\_relation\_2Etransitive}~\\
A\_27a)~V11r)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Etransitive}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~\\
A\_27a~A\_27a)~V11r)))){\unicdcdcdhd}(((p~(ap~(\mname{c\_2Eset\_\_relation\_2Eantisym}~\\
A\_27a)~V11r)){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2Eantisymmetric}~A\_27a)~(ap~\\
(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27a)~V11r)))){\unicdcdcdhd}(\\
((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Epartial\_\_order}~A\_27a)~V11r)~\\
(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2EWeakOrder}~\\
A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27a)~V11r)))){\unicdcdcdhd}\\
(((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}~A\_27a)~V11r)~\\
(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2EWeakLinearOrder}~\\
A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27a)~V11r)))){\unicdcdcdhd}\\
((p~(ap~(ap~(\mname{c\_2Eset\_\_relation\_2Estrict\_\_linear\_\_order}~A\_27a)~\\
V11r)~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))){\unicdbdjded}(p~(ap~(\mname{c\_2Erelation\_2EStrongLinearOrder}~\\
A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Ereln\_\_to\_\_rel}~A\_27a~A\_27a)~V11r))))))))))))))))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
