\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Ewellorder\_2Ewellorder\_\_fromNat\_\_SUM (TMF4GDdMkq4a94z6vCcjST5hDoGefJnWSn5)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengbgchlgfgchpfcdfebecgcgchfggh}
We define $\mname{c\_2Emarker\_2EAbbrev}$ to be ${{\uniddcgcg} V0x \in {2} .  {V0x}}$.
\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}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
\begin{definition}\label{def: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: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\_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 $ c\_2Esum\_2EABS\_\_sum : {{\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\_2Esum\_2EABS\_\_sum~\\
A\_27a~A\_27b \in ((ty\_2Esum\_2Esum~A\_27a~A\_27b)^{{({{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}^{{2}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfcdfebecedfpfpfdhfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhfhngpfcdfejeoecf}
We define $\mname{c\_2Esum\_2EINR}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0e \in {A\_27b} .  {({{{ap}~{({{{c\_2Esum\_2EABS\_\_sum}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V1b \in {2} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3y}})}}~{V0e}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{V1b}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfhngpfcdfejeoeme}
We define $\mname{c\_2Esum\_2EINL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {({{{ap}~{({{{c\_2Esum\_2EABS\_\_sum}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V1b \in {2} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}}~{V0e}})}})}}~{V1b}})}})}})}})}})}}}}$.
\end{definition}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfdddepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3C\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\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: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:dgpfcdfeahbgjgchpfcdfepfcdddpfcddd}
We define $\mname{c\_2Epair\_2E\_23\_23}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} A\_27d : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27c}^{{A\_27a}}})} .  {{\uniddcgcg} V1g \in {({{A\_27d}^{{A\_27b}}})} .  {{\uniddcgcg} V2p \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27c}}~{A\_27d}})}}~{({{{ap}~{V0f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27a}}~{A\_27b}})}}~{V2p}})}})}})}}~{({{{ap}~{V1g}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{A\_27b}})}}~{V2p}})}})}})}}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejenebehefe}
We define $\mname{c\_2Epred\_\_set\_2EIMAGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27b}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{2}})}}~{({{{ap}~{V0f}}~{V2x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}})}})}}}}}$.
\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: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:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoeke}
We define $\mname{c\_2Epred\_\_set\_2EINJ}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2t \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V3x}})}})}}~{V2t}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V4x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V5y}})}}~{V1s}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V4x}})}})}}~{({{{ap}~{V0f}}~{V5y}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfechfgggmgfgihjgghfg}
We define $\mname{c\_2Eset\_\_relation\_2Ereflexive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V2x}})}})}}~{V0r}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfebgogehjgdhjhng}
We define $\mname{c\_2Eset\_\_relation\_2Eantisym}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2y}})}}~{V1x}})}})}}~{V0r}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfeehchbgogdhjgehjgghfg}
We define $\mname{c\_2Eset\_\_relation\_2Etransitive}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2y}})}}~{V3z}})}})}}~{V0r}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V3z}})}})}}~{V0r}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfedfffcedffeef}
We define $\mname{c\_2Epred\_\_set\_2ESUBSET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfemgjgogfgbgchpfpfpgchegfgch}
We define $\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0r}})}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0r}})}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Etransitive}}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Eantisym}}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3y}})}}~{V1s}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V3y}})}})}}~{V0r}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V3y}})}}~{V2x}})}})}}~{V0r}})}})}})}})}})}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfgehpfpfchfgmgbgehjgpgogpfcdfedhehchjgdgeh}
We define $\mname{c\_2Eset\_\_relation\_2Estrict}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{A\_27a}}~{A\_27a}}~{({{{ty\_2Epair\_2Eprod}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}}~{V0r}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgggpgfhogegfgeg}
We define $\mname{c\_2Ewellorder\_2Ewellfounded}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2w \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2w}})}}~{V1s}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3min \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3min}})}}~{V1s}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4w \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V4w}})}}~{V3min}})}})}}~{V0R}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V4w}})}}~{V1s}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgch}
We define $\mname{c\_2Ewellorder\_2Ewellorder}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{2}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ewellorder\_2Ewellfounded}}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Estrict}}~{A\_27a}})}}~{V0R}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Elinear\_\_order}}~{A\_27a}})}}~{V0R}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eset\_\_relation\_2Ereflexive}}~{A\_27a}})}}~{V0R}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{A\_27a}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Edomain}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}})}}~{({{{ap}~{({{{\mname{c\_2Eset\_\_relation\_2Erange}}~{A\_27a}}~{A\_27a}})}}~{V0R}})}})}})}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} 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} 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} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  (({\unicdcdaddd} V2x \in A\_27a .  ((V2x=V1a){\unicdcdcdhd}(p~(ap~V0P~V2x)))){\unicdbdjded}(p~(\\
ap~V0P~V1a)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeafbejecfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedemepedffeeepfpfafbejecfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~A\_27b)~(ap~(c\_2Epair\_2EFST~A\_27a~A\_27b)~V0x))~(ap~(c\_2Epair\_2ESND~\\
A\_27a~A\_27b)~V0x))=V0x))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeafbejecf}
\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} V0f \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1x \in \\
A\_27a .  ({\unicdcdadad} V2y \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~\\
A\_27b~A\_27c)~V0f)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V1x)~V2y))=\\
(ap~(ap~V0f~V1x)~V2y)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeffoedeffcfcfjfpfpfeefege}
\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} V0P \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  (({\unicdcdaddd} V1p \in \\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  (p~(ap~V0P~V1p))){\unicdbdjded}({\unicdcdaddd} V2p\_\_1 \in \\
A\_27a .  ({\unicdcdaddd} V3p\_\_2 \in A\_27b .  (p~(ap~V0P~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~A\_27b)~V2p\_\_1)~V3p\_\_2)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfefeifjedfefdfpfpfafcfpeee}
\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} V0f \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V1g \in (A\_27d^{{A\_27c}}) .  ({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3y \in \\
A\_27c .  ((ap~(ap~(ap~(\mname{c\_2Epair\_2E\_23\_23}~A\_27a~A\_27c~A\_27b~A\_27d)~\\
V0f)~V1g)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27c)~V2x)~V3y))=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27b~A\_27d)~(ap~V0f~V2x))~(ap~V1g~V3y)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeafbejecfpfpfnebeafpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ((V0s=V1t){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V0s)){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfefeifeffeoedfjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in ((ty\_2Epair\_2Eprod~A\_27a~2)^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1v)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b)~V0f))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~2)~V1v)~\mname{c\_2Ebool\_2ET})=(ap~V0f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffedejegejedebeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V2x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0y \in A\_27b .  ({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V2f \in (A\_27b^{{A\_27a}}) .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~V0y)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~\\
A\_27a~A\_27b)~V2f)~V1s))){\unicdbdjded}({\unicdcdaddd} V3x \in A\_27a .  ((V0y=(ap~V2f~V3x)){\unicdcdcdhd}\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V3x)~V1s))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfjenebehefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V0x)~V1s)){\unicdbdjdcd}({\unicdcdadad} V2f \in (A\_27b^{{A\_27a}}) .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27b)~(ap~V2f~V0x))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~A\_27a~A\_27b)~\\
V2f)~V1s)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejenebehefepfpfjeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
({\unicdcdadad} V0y \in A\_27a .  ({\unicdcdadad} V1x \in A\_27a .  (((ap~(\mname{c\_2Esum\_2EINL}~\\
A\_27a~A\_27b)~V1x)=(ap~(\mname{c\_2Esum\_2EINL}~A\_27a~A\_27b)~V0y)){\unicdbdjded}(V1x=\\
V0y)))){\unicdcdcdhd}({\unicdcdadad} V2y \in A\_27b .  ({\unicdcdadad} V3x \in A\_27b .  (((ap~(\mname{c\_2Esum\_2EINR}~\\
A\_27a~A\_27b)~V3x)=(ap~(\mname{c\_2Esum\_2EINR}~A\_27a~A\_27b)~V2y)){\unicdbdjded}(V3x=\\
V2y)))))\end{array}
\label{hyp:ehigngpfcdfedhfhngpfcdfejeoecfpfpfjeoemepfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(\mname{c\_2Ewellorder\_2Ewellorder}~\\
ty\_2Enum\_2Enum)~(ap~(c\_2Epred\_\_set\_2EGSPEC~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum))~(ap~(\mname{c\_2Epair\_2EUNCURRY}~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum~\\
(ty\_2Epair\_2Eprod~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~\\
2))~({\uniddcgcg} V1i \in ty\_2Enum\_2Enum .  ({\uniddcgcg} V2j \in ty\_2Enum\_2Enum .  (ap~\\
(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~\\
2)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~V1i)~\\
V2j))~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1i)~V2j))~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V2j)~V0n))))))))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgchpfpfggchpgngoebgeh}
\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\_27a}})}}) .  ({\unicdcdadad} V1f \in \\
(A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V2t \in (2^{{A\_27b}}) .  (((p~(ap~(\mname{c\_2Ewellorder\_2Ewellorder}~\\
A\_27a)~V0r)){\unicdcdcdhd}(p~(ap~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINJ}~A\_27a~A\_27b)~V1f)~\\
(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~A\_27a)~(ap~(\mname{c\_2Eset\_\_relation\_2Edomain}~\\
A\_27a~A\_27a)~V0r))~(ap~(\mname{c\_2Eset\_\_relation\_2Erange}~A\_27a~A\_27a)~\\
V0r)))~V2t))){\unicdbdjdcd}(p~(ap~(\mname{c\_2Ewellorder\_2Ewellorder}~A\_27b)~(ap~(ap~\\
(\mname{c\_2Epred\_\_set\_2EIMAGE}~(ty\_2Epair\_2Eprod~A\_27a~A\_27a)~(ty\_2Epair\_2Eprod~\\
A\_27b~A\_27b))~(ap~(ap~(\mname{c\_2Epair\_2E\_23\_23}~A\_27a~A\_27a~A\_27b~A\_27b)~\\
V1f)~V1f))~V0r)))))))\end{array}
\label{hyp:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfejeoekepfpfahchfgdhfgchghfgdhpfpfhhfgmgmgpgchegfgch}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfehhfgmgmgpgchegfgchpfcdfehhfgmgmgpgchegfgchpfpfggchpgngoebgehpfpfdfffne}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\\
p~(ap~(\mname{c\_2Ewellorder\_2Ewellorder}~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~\\
A\_27a))~(ap~(c\_2Epred\_\_set\_2EGSPEC~(ty\_2Epair\_2Eprod~(ty\_2Esum\_2Esum~\\
ty\_2Enum\_2Enum~A\_27a)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~A\_27a))~\\
(ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum))~(ap~(\mname{c\_2Epair\_2EUNCURRY}~\\
ty\_2Enum\_2Enum~ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~(ty\_2Epair\_2Eprod~\\
(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~A\_27a)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~\\
A\_27a))~2))~({\uniddcgcg} V1i \in ty\_2Enum\_2Enum .  ({\uniddcgcg} V2j \in ty\_2Enum\_2Enum .  \\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Epair\_2Eprod~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~\\
A\_27a)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~A\_27a))~2)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~A\_27a)~(ty\_2Esum\_2Esum~ty\_2Enum\_2Enum~\\
A\_27a))~(ap~(\mname{c\_2Esum\_2EINL}~ty\_2Enum\_2Enum~A\_27a)~V1i))~(ap~(\mname{c\_2Esum\_2EINL}~\\
ty\_2Enum\_2Enum~A\_27a)~V2j)))~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~(ap~(ap~\\
\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1i)~V2j))~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V2j)~V0n))))))))))\end{array}$$
\end{theorem}
\end{document}
