\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\_2Ecardinal\_2Elist\_\_SING (TMJPhGuGVzHDcguwkcTqFmhUap6PdnivcPV)}
\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: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: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: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}
\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:dgpfcdfeahchfgegpfpfdhfgehpfcdfedfffcfke}
We define $\mname{c\_2Epred\_\_set\_2ESURJ}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2t \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V3x}})}})}}~{V2t}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V4x \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{V4x}})}}~{V2t}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V5y}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V5y}})}})}}~{V4x}})}})}})}})}})}})}})}})}}}}}}$.
\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:dgpfcdfeahchfgegpfpfdhfgehpfcdfecejeke}
We define $\mname{c\_2Epred\_\_set\_2EBIJ}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2t \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epred\_\_set\_2EINJ}}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V1s}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epred\_\_set\_2ESURJ}}~{A\_27a}}~{A\_27b}})}}~{V0f}})}}~{V1s}})}}~{V2t}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgbgchegjgogbgmgpfcdfedgbgchegfgbh}
We define $\mname{c\_2Ecardinal\_2Ecardeq}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0s1 \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1s2 \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epred\_\_set\_2EBIJ}}~{A\_27a}}~{A\_27b}})}}~{V2f}})}}~{V0s1}})}}~{V1s2}})}})}})}}}}}$.
\end{definition}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ c\_2Elist\_2ELIST\_\_TO\_\_SET : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a \in \\
((2^{{A\_27a}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemejedfefpfpfefpepfpfdffeef}
\end{equation}
Let $ ty\_2Epair\_2Eprod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_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}
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:dgpfcdfedgbgchegjgogbgmgpfcdfemgjgdheh}
We define $\mname{c\_2Ecardinal\_2Elist}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0A \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{\uniddcgcg} V1l \in {({{ty\_2Elist\_2Elist}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}}~{2}})}}~{V1l}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2e \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2e}})}}~{({{{ap}~{({{c\_2Elist\_2ELIST\_\_TO\_\_SET}~{A\_27a}})}}~{V1l}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2e}})}}~{V0A}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfele}
We define $\mname{c\_2Ecombin\_2EK}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1y \in {A\_27b} .  {V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\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\_2Elist\_2ELENGTH : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ELENGTH~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemefeoeheefie}
\end{equation}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\end{definition}
Let $ c\_2Elist\_2ECONS : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ECONS~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedepeoedf}
\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}
Let $ c\_2Elist\_2EGENLIST\_\_AUX : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EGENLIST\_\_AUX~A\_27a \in \\
((((ty\_2Elist\_2Elist~A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{ty\_2Enum\_2Enum}})^{{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfehefeoemejedfefpfpfbeffif}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\end{equation}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\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: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}
Let $ c\_2Elist\_2EGENLIST : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EGENLIST~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{ty\_2Enum\_2Enum}})^{{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfehefeoemejedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfefeneafefjf}
We define $\mname{c\_2Epred\_\_set\_2EEMPTY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2EF}}})}}$.
\end{definition}
\begin{definition}\label{def: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: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:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoedffecfef}
We define $\mname{c\_2Epred\_\_set\_2EINSERT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2y}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2y}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2y}})}}~{V1s}})}})}})}})}})}}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
\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{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} 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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
2 .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdbdjdcd}(p~V1Q))){\unicdbdjded}(({\unicdcdaddd} V3x \in \\
A\_27a .  (p~(ap~V0P~V3x))){\unicdbdjdcd}(p~V1Q)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfgepecfbememepfpfjeneafpfpfefiene}
\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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27a~A\_27b)~V0x)~V1y)=V0x)))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfelepfpfefiene}
\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} A\_27e .  nonempty~\\
A\_27e{\unicdbdjdcd}{\unicdcdadad} A\_27f .  nonempty~A\_27f{\unicdbdjdcd}(({\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V1v \in A\_27c .  ((ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~A\_27c~A\_27b)~\\
(ap~(\mname{c\_2Ecombin\_2EK}~A\_27c~A\_27b)~V1v))~V0f)=(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27c~A\_27a)~V1v)))){\unicdcdcdhd}({\unicdcdadad} V2f \in (A\_27e^{{A\_27d}}) .  ({\unicdcdadad} V3v \in \\
A\_27d .  ((ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27f~A\_27e~A\_27d)~V2f)~(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27d~A\_27f)~V3v))=(ap~(\mname{c\_2Ecombin\_2EK}~A\_27e~A\_27f)~(ap~V2f~V3v))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfelepfpfpgpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(((ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
(c\_2Elist\_2ENIL~A\_27a))=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}({\unicdcdadad} V0h \in A\_27a .  (\\
{\unicdcdadad} V1t \in (ty\_2Elist\_2Elist~A\_27a) .  ((ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V0h)~V1t))=(ap~\mname{c\_2Enum\_2ESUC}~\\
(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V1t))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemefeoeheefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}}) .  \\
(((p~(ap~V0P~(c\_2Elist\_2ENIL~A\_27a))){\unicdcdcdhd}({\unicdcdadad} V1t \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((p~(ap~V0P~V1t)){\unicdbdjdcd}({\unicdcdadad} V2h \in A\_27a .  (p~(ap~V0P~(ap~(ap~(\\
c\_2Elist\_2ECONS~A\_27a)~V2h)~V1t))))))){\unicdbdjdcd}({\unicdcdadad} V3l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  (p~(ap~V0P~V3l)))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemgjgdhehpfpfjgogegfhdgehjgpgog}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a0 \in A\_27a .  ({\unicdcdadad} V1a1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2a0\_27 \in A\_27a .  ({\unicdcdadad} V3a1\_27 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  (((ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V0a0)~\\
V1a1)=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V2a0\_27)~V3a1\_27)){\unicdbdjded}((V0a0=\\
V2a0\_27){\unicdcdcdhd}(V1a1=V3a1\_27)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfedepeoedfpfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0x \in A\_27a .  ((p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~\\
(c\_2Elist\_2ENIL~A\_27a)))){\unicdbdjded}False)){\unicdcdcdhd}({\unicdcdadad} V1x \in A\_27a .  ({\unicdcdadad} V2h \in \\
A\_27a .  ({\unicdcdadad} V3t \in (ty\_2Elist\_2Elist~A\_27a) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V1x)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2h)~V3t)))){\unicdbdjded}((V1x=V2h){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V1x)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~A\_27a)~V3t)))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenefene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27a^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((ap~(c\_2Elist\_2ELENGTH~A\_27a)~\\
(ap~(ap~(c\_2Elist\_2EGENLIST~A\_27a)~V0f)~V1n))=V1n)))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemefeoeheefiepfpfhefeoemejedfef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27a^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((ap~(ap~(c\_2Elist\_2EGENLIST~A\_27a)~\\
V0f)~(ap~\mname{c\_2Enum\_2ESUC}~V1n))=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~(\\
ap~V0f~\mname{c\_2Enum\_2E0}))~(ap~(ap~(c\_2Elist\_2EGENLIST~A\_27a)~(ap~(ap~\\
(\mname{c\_2Ecombin\_2Eo}~ty\_2Enum\_2Enum~A\_27a~ty\_2Enum\_2Enum)~V0f)~\mname{c\_2Enum\_2ESUC}))~\\
V1n)))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfehefeoemejedfefpfpfdepeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27a^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (((ap~(ap~(c\_2Elist\_2EGENLIST~A\_27a)~\\
V0f)~\mname{c\_2Enum\_2E0})=(c\_2Elist\_2ENIL~A\_27a)){\unicdcdcdhd}((ap~(ap~(c\_2Elist\_2EGENLIST~\\
A\_27a)~V0f)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V1n))=(ap~(ap~(ap~(c\_2Elist\_2EGENLIST\_\_AUX~\\
A\_27a)~V0f)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V1n))~(c\_2Elist\_2ENIL~\\
A\_27a))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfehefeoemejedfefpfpfoeffnefecfbemedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1f \in \\
(A\_27a^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(ap~(c\_2Elist\_2ELIST\_\_TO\_\_SET~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2EGENLIST~A\_27a)~V1f)~V2n)))){\unicdbdjded}({\unicdcdaddd} V3m \in \\
ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V3m)~V2n)){\unicdcdcdhd}(V0x=\\
(ap~V1f~V3m))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenefenepfpfhefeoemejedfef}
\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} V0f \in ((ty\_2Epair\_2Eprod~A\_27a~2)^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1v)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b)~V0f))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~2)~V1v)~\mname{c\_2Ebool\_2ET})=(ap~V0f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffedejegejedebeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (\preoh (p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfeoepeefpfpfjeoepfpffeneafefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ({\unicdcdadad} V2s \in (2^{{A\_27a}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V0x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V1y)~V2s))){\unicdbdjded}((V0x=\\
V1y){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~V2s)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfjeoedffecfef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V2t \in \\
(2^{{A\_27b}}) .  ((p~(ap~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EBIJ}~A\_27a~A\_27b)~V0f)~\\
V1s)~V2t)){\unicdbdjded}(({\unicdcdadad} V3x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V3x)~V1s)){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~(ap~V0f~V3x))~V2t)))){\unicdcdcdhd}\\
({\unicdcdaddd} V4g \in (A\_27a^{{A\_27b}}) .  (({\unicdcdadad} V5x \in A\_27b .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27b)~V5x)~V2t)){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~(ap~V4g~V5x))~\\
V1s)))){\unicdcdcdhd}(({\unicdcdadad} V6x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V6x)~V1s)){\unicdbdjdcd}((ap~V4g~(ap~V0f~V6x))=V6x))){\unicdcdcdhd}({\unicdcdadad} V7x \in A\_27b .  (\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~V7x)~V2t)){\unicdbdjdcd}((ap~V0f~(ap~V4g~V7x))=\\
V7x)))))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfecejekepfpfjegegepfpfjeoegf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedgbgchegjgogbgmgpfcdfemgjgdhehpfpfdfjeoehe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0e \in A\_27a .  (p~(ap~(ap~(\mname{c\_2Ecardinal\_2Ecardeq}~\\
(ty\_2Elist\_2Elist~A\_27a)~ty\_2Enum\_2Enum)~(ap~(\mname{c\_2Ecardinal\_2Elist}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~A\_27a)~V0e)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
A\_27a))))~(\mname{c\_2Epred\_\_set\_2EUNIV}~ty\_2Enum\_2Enum))))\end{array}$$
\end{theorem}
\end{document}
