\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\_2Ebag\_2EBAG\_\_IMAGE\_\_FINITE\_\_RESTRICTED\_\_I (TMM8r1ZzYUopRj3EgcKhaJHz3m73BG6CPZ6)}
\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}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\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: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:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpfgejemeeffecf}
We define $\mname{c\_2Ebag\_2EBAG\_\_FILTER}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{\uniddcgcg} V2e \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{V0P}}~{V2e}})}})}}~{({{{ap}~{V1b}}~{V2e}})}})}}~{\mname{c\_2Enum\_2E0}}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
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}
\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\_2Earithmetic\_2E\_2B : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2B \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdce}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefbd}
We define $\mname{c\_2Earithmetic\_2EBIT1}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
Let $ c\_2Ebag\_2EBAG\_\_CARD : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebag\_2EBAG\_\_CARD~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgbghgpfcdfecebehepfpfdebecfee}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpfjeoedffecfef}
We define $\mname{c\_2Ebag\_2EBAG\_\_INSERT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {{\uniddcgcg} V1b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}}~{V0e}})}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{({{{ap}~{V1b}}~{V0e}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{V1b}}~{V2x}})}})}})}}}}$.
\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:dgpfcdfecgbghgpfcdfefeneafefjfpfpfcebehe}
We define $\mname{c\_2Ebag\_2EEMPTY\_\_BAG}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{ty\_2Enum\_2Enum}}~{A\_27a}})}}~{\mname{c\_2Enum\_2E0}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfegejeoejeeffepfpfcebehe}
We define $\mname{c\_2Ebag\_2EFINITE\_\_BAG}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V1P \in {({{2}^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1P}}~{({{\mname{c\_2Ebag\_2EEMPTY\_\_BAG}}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V1P}}~{V2b}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3e \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_INSERT}}~{A\_27a}})}}~{V3e}})}}~{V2b}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{V1P}}~{V0b}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfemefeef}
We define $\mname{c\_2Ebool\_2ELET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{V0f}}~{V1x}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpfjenebehefe}
We define $\mname{c\_2Ebag\_2EBAG\_\_IMAGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27a}^{{A\_27b}}})} .  {{\uniddcgcg} V1b \in {({{ty\_2Enum\_2Enum}^{{A\_27b}}})} .  {({{\uniddcgcg} V2e \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Ebool\_2ELET}}~{({{ty\_2Enum\_2Enum}^{{A\_27b}}})}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3sb \in {({{ty\_2Enum\_2Enum}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{\mname{c\_2Ebag\_2EFINITE\_\_BAG}}~{A\_27b}})}}~{V3sb}})}})}}~{({{{ap}~{({{c\_2Ebag\_2EBAG\_\_CARD}~{A\_27b}})}}~{V3sb}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_FILTER}}~{A\_27b}})}}~{({{\uniddcgcg} V4e0 \in {A\_27b} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{V0f}}~{V4e0}})}})}}~{V2e}})}})}})}}~{V1b}})}})}})}}}}}$.
\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: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: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:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfe}
We define $\mname{c\_2Earithmetic\_2E\_3E}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V1n}})}}~{V0m}})}}}$.
\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:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3E\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3E}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpfjeoeoe}
We define $\mname{c\_2Ebag\_2EBAG\_\_INN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V2b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3E\_3D}}}~{({{{ap}~{V2b}}~{V0e}})}})}}~{V1n}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpfjeoe}
We define $\mname{c\_2Ebag\_2EBAG\_\_IN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {{\uniddcgcg} V1b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_INN}}~{A\_27a}})}}~{V0e}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V1b}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgbghgpfcdfecebehepfpffegffecfjf}
We define $\mname{c\_2Ebag\_2EBAG\_\_EVERY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1b \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2e \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebag\_2EBAG\_\_IN}}~{A\_27a}})}}~{V2e}})}}~{V1b}})}})}}~{({{{ap}~{V0P}}~{V2e}})}})}})}})}}}}$.
\end{definition}
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} V2b \in (ty\_2Enum\_2Enum^{{A\_27a}}) .  (((ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_INSERT}~\\
A\_27a)~V0x)~V2b)=(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_INSERT}~A\_27a)~V1y)~V2b)){\unicdbdjded}\\
(V0x=V1y)))))\end{array}
\label{hyp:ehigngpfcdfecgbghgpfcdfedepfpfcebehepfpfjeoedffecfefpfpfpeoefepfpfpeoefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}}) .  \\
(((p~(ap~V0P~(\mname{c\_2Ebag\_2EEMPTY\_\_BAG}~A\_27a))){\unicdcdcdhd}({\unicdcdadad} V1b \in (ty\_2Enum\_2Enum^{{A\_27a}}) .  \\
(((p~(ap~(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~A\_27a)~V1b)){\unicdcdcdhd}(p~(ap~V0P~V1b))){\unicdbdjdcd}\\
({\unicdcdadad} V2e \in A\_27a .  (p~(ap~V0P~(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_INSERT}~A\_27a)~\\
V2e)~V1b))))))){\unicdbdjdcd}({\unicdcdadad} V3b \in (ty\_2Enum\_2Enum^{{A\_27a}}) .  ((p~(ap~\\
(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~A\_27a)~V3b)){\unicdbdjdcd}(p~(ap~V0P~V3b))))))\end{array}
\label{hyp:ehigngpfcdfecgbghgpfcdfedfefcfpeoehepfpfgejeoejeeffepfpfcebehepfpfjeoeeeffdeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}((p~(ap~(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~\\
A\_27a)~(\mname{c\_2Ebag\_2EEMPTY\_\_BAG}~A\_27a))){\unicdcdcdhd}({\unicdcdadad} V0e \in A\_27a .  (\\
{\unicdcdadad} V1b \in (ty\_2Enum\_2Enum^{{A\_27a}}) .  ((p~(ap~(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_INSERT}~A\_27a)~V0e)~V1b))){\unicdbdjded}(p~(ap~\\
(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~A\_27a)~V1b))))))\end{array}
\label{hyp:ehigngpfcdfecgbghgpfcdfegejeoejeeffepfpfcebehepfpfefiene}
\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} V0b \in (ty\_2Enum\_2Enum^{{A\_27a}}) .  ({\unicdcdadad} V1f \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V2e \in A\_27a .  ((p~(ap~(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~A\_27a)~V0b)){\unicdbdjdcd}\\
((ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_IMAGE}~A\_27b~A\_27a)~V1f)~(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_INSERT}~\\
A\_27a)~V2e)~V0b))=(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_INSERT}~A\_27b)~(ap~V1f~\\
V2e))~(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_IMAGE}~A\_27b~A\_27a)~V1f)~V0b)))))))\end{array}
\label{hyp:ehigngpfcdfecgbghgpfcdfecebehepfpfjenebehefepfpfgejeoejeeffepfpfjeoedffecfef}
\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} V0b \in (ty\_2Enum\_2Enum^{{A\_27a}}) .  ({\unicdcdadad} V1f \in (A\_27b^{{A\_27a}}) .  \\
((p~(ap~(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~A\_27a)~V0b)){\unicdbdjdcd}(((ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_IMAGE}~\\
A\_27b~A\_27a)~V1f)~V0b)=(\mname{c\_2Ebag\_2EEMPTY\_\_BAG}~A\_27b)){\unicdbdjded}(V0b=(\\
\mname{c\_2Ebag\_2EEMPTY\_\_BAG}~A\_27a))))))\end{array}
\label{hyp:ehigngpfcdfecgbghgpfcdfecebehepfpfjenebehefepfpffebfpfpffeneafefjf}
\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^{{A\_27a}}) .  (p~(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_EVERY}~A\_27a)~\\
V0P)~(\mname{c\_2Ebag\_2EEMPTY\_\_BAG}~A\_27a)))){\unicdcdcdhd}({\unicdcdadad} V1P \in (2^{{A\_27b}}) .  \\
({\unicdcdadad} V2e \in A\_27b .  ({\unicdcdadad} V3b \in (ty\_2Enum\_2Enum^{{A\_27b}}) .  ((p~\\
(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_EVERY}~A\_27b)~V1P)~(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_INSERT}~\\
A\_27b)~V2e)~V3b))){\unicdbdjded}((p~(ap~V1P~V2e)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_EVERY}~\\
A\_27b)~V1P)~V3b))))))))\end{array}
\label{hyp:ehigngpfcdfecgbghgpfcdfecebehepfpffegffecfjfpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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{\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}
\begin{theorem}\label{thm:ehigngpfcdfecgbghgpfcdfecebehepfpfjenebehefepfpfgejeoejeeffepfpfcffedfefcfjedeeffeeepfpfje}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (A\_27a^{{A\_27a}}) .  ({\unicdcdadad} V1b \in \\
(ty\_2Enum\_2Enum^{{A\_27a}}) .  (((p~(ap~(\mname{c\_2Ebag\_2EFINITE\_\_BAG}~A\_27a)~\\
V1b)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ebag\_2EBAG\_\_EVERY}~A\_27a)~({\uniddcgcg} V2e \in A\_27a .  \\
(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27a)~(ap~V0f~V2e))~V2e)))~V1b))){\unicdbdjdcd}((ap~(\\
ap~(\mname{c\_2Ebag\_2EBAG\_\_IMAGE}~A\_27a~A\_27a)~V0f)~V1b)=V1b))))\end{array}$$
\end{theorem}
\end{document}
