\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\_2Eind\_\_type\_2EZCONSTR\_\_ZBOT (TMe12ayAq6BNbRVdDVBhyx2tpbjQb5NAkKr)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def: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: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\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Eind\_\_type\_2ENUMSND : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Eind\_\_type\_2ENUMSND} \in {({{ty\_2Enum\_2Enum}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogegpfpfehjhahfgpfcdfeoeffnedfoeee}
\end{equation}
Let $ c\_2Eind\_\_type\_2ENUMFST : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Eind\_\_type\_2ENUMFST} \in {({{ty\_2Enum\_2Enum}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogegpfpfehjhahfgpfcdfeoeffnegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogegpfpfehjhahfgpfcdfejeoekege}
We define $\mname{c\_2Eind\_\_type\_2EINJF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0f \in {({{({{({{2}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})}^{{ty\_2Enum\_2Enum}}})} .  {({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{V0f}}~{({{{ap}~{c\_2Eind\_\_type\_2ENUMFST}}~{V1n}})}})}}~{({{{ap}~{c\_2Eind\_\_type\_2ENUMSND}}~{V1n}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogegpfpfehjhahfgpfcdfejeoekebe}
We define $\mname{c\_2Eind\_\_type\_2EINJA}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0a \in {A\_27a} .  {({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2b}})}}~{V0a}})}})}})}}}$.
\end{definition}
Let $ c\_2Eind\_\_type\_2ENUMRIGHT : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Eind\_\_type\_2ENUMRIGHT} \in {({{ty\_2Enum\_2Enum}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogegpfpfehjhahfgpfcdfeoeffnecfjeheieef}
\end{equation}
Let $ c\_2Eind\_\_type\_2ENUMLEFT : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Eind\_\_type\_2ENUMLEFT} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogegpfpfehjhahfgpfcdfeoeffnemefegeef}
\end{equation}
\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:dgpfcdfejgogegpfpfehjhahfgpfcdfejeoekeaf}
We define $\mname{c\_2Eind\_\_type\_2EINJP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0f1 \in {({{({{2}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})} .  {{\uniddcgcg} V1f2 \in {({{({{2}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})} .  {({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V3a \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{2}})}}~{({{{ap}~{c\_2Eind\_\_type\_2ENUMLEFT}}~{V2n}})}})}}~{({{{ap}~{({{{ap}~{V0f1}}~{({{{ap}~{c\_2Eind\_\_type\_2ENUMRIGHT}}~{V2n}})}})}}~{V3a}})}})}}~{({{{ap}~{({{{ap}~{V1f2}}~{({{{ap}~{c\_2Eind\_\_type\_2ENUMRIGHT}}~{V2n}})}})}}~{V3a}})}})}})}})}}}}$.
\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}
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:dgpfcdfejgogegpfpfehjhahfgpfcdfejeoekeoe}
We define $\mname{c\_2Eind\_\_type\_2EINJN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{\uniddcgcg} V2a \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V1n}})}}~{V0m}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogegpfpfehjhahfgpfcdfekfdepeoedfefcf}
We define $\mname{c\_2Eind\_\_type\_2EZCONSTR}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0c \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1i \in {A\_27a} .  {{\uniddcgcg} V2r \in {({{({{({{2}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Eind\_\_type\_2EINJP}}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Eind\_\_type\_2EINJN}}~{A\_27a}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V0c}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Eind\_\_type\_2EINJP}}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Eind\_\_type\_2EINJA}}~{A\_27a}})}}~{V1i}})}})}}~{({{{ap}~{({{\mname{c\_2Eind\_\_type\_2EINJF}}~{A\_27a}})}}~{V2r}})}})}})}}}}}$.
\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}
\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:dgpfcdfejgogegpfpfehjhahfgpfcdfekfcepeef}
We define $\mname{c\_2Eind\_\_type\_2EZBOT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Eind\_\_type\_2EINJP}}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Eind\_\_type\_2EINJN}}~{A\_27a}})}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{2}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V0z \in {({{({{2}^{{A\_27a}}})}^{{ty\_2Enum\_2Enum}}})} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\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}
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 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n1 \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V1n2 \in ty\_2Enum\_2Enum .  (((ap~(\mname{c\_2Eind\_\_type\_2EINJN}~A\_27a)~\\
V0n1)=(ap~(\mname{c\_2Eind\_\_type\_2EINJN}~A\_27a)~V1n2)){\unicdbdjded}(V0n1=V1n2))))\end{array}
\label{hyp:ehigngpfcdfejgogegpfpfehjhahfgpfcdfejeoekeoepfpfjeoeke}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f1 \in ((2^{{A\_27a}})^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V1f1\_27 \in ((2^{{A\_27a}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V2f2 \in \\
((2^{{A\_27a}})^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V3f2\_27 \in ((2^{{A\_27a}})^{{ty\_2Enum\_2Enum}}) .  \\
(((ap~(ap~(\mname{c\_2Eind\_\_type\_2EINJP}~A\_27a)~V0f1)~V2f2)=(ap~(ap~(\mname{c\_2Eind\_\_type\_2EINJP}~\\
A\_27a)~V1f1\_27)~V3f2\_27)){\unicdbdjded}((V0f1=V1f1\_27){\unicdcdcdhd}(V2f2=V3f2\_27)))))))\end{array}
\label{hyp:ehigngpfcdfejgogegpfpfehjhahfgpfcdfejeoekeafpfpfjeoeke}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\preoh ((ap~\mname{c\_2Enum\_2ESUC}~V0n)=\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfeogfhngpfcdfeoepeefpfpfdfffde}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfejgogegpfpfehjhahfgpfcdfekfdepeoedfefcfpfpfkfcepeef}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0c \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V1i \in A\_27a .  ({\unicdcdadad} V2r \in (((2^{{A\_27a}})^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}}) .  \\
(\preoh ((ap~(ap~(ap~(\mname{c\_2Eind\_\_type\_2EZCONSTR}~A\_27a)~V0c)~V1i)~V2r)=\\
(\mname{c\_2Eind\_\_type\_2EZBOT}~A\_27a))))))\end{array}$$
\end{theorem}
\end{document}
