\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\_2Elist\_2ESINGL\_\_APPLY\_\_MAP (TMG7B443g7JrzjQVNg5eL2fPdHBoxr37kmx)}
\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}
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\_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\_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\_2Elist\_2EMAP : {{\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\_2Elist\_2EMAP~\\
A\_27a~A\_27b \in (((ty\_2Elist\_2Elist~A\_27b)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{A\_27b}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfenebeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfede}
We define $\mname{c\_2Ecombin\_2EC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27b} .  {({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2y}})}}~{V1x}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Elist\_2EFLAT : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EFLAT~A\_27a \in ((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfegemebeef}
\end{equation}
\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:dgpfcdfemgjgdhehpfcdfemejedfefpfpfcejeoeee}
We define $\mname{c\_2Elist\_2ELIST\_\_BIND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0l \in {({{ty\_2Elist\_2Elist}~{A\_27b}})} .  {{\uniddcgcg} V1f \in {({{({{ty\_2Elist\_2Elist}~{A\_27a}})}^{{A\_27b}}})} .  {({{{ap}~{({{c\_2Elist\_2EFLAT}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{c\_2Elist\_2EMAP}~{A\_27b}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{V1f}})}}~{V0l}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgjgdhehpfcdfemejedfefpfpfbeafafmejf}
We define $\mname{c\_2Elist\_2ELIST\_\_APPLY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0fs \in {({{ty\_2Elist\_2Elist}~{({{A\_27a}^{{A\_27b}}})}})} .  {{\uniddcgcg} V1xs \in {({{ty\_2Elist\_2Elist}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Elist\_2ELIST\_\_BIND}}~{A\_27a}}~{({{A\_27a}^{{A\_27b}}})}})}}~{V0fs}})}}~{({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2EC}}~{({{A\_27a}^{{A\_27b}}})}}~{({{ty\_2Elist\_2Elist}~{A\_27b}})}}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}})}}~{({{{c\_2Elist\_2EMAP}~{A\_27b}}~{A\_27a}})}})}}~{V1xs}})}})}}}}}$.
\end{definition}
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} 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} 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\_27b .  ({\unicdcdadad} V2y \in A\_27a .  ((ap~(ap~(ap~(\mname{c\_2Ecombin\_2EC}~A\_27a~A\_27b~\\
A\_27c)~V0f)~V1x)~V2y)=(ap~(ap~V0f~V2y)~V1x)))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfedepfpfefiene}
\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\_27b .  ({\unicdcdadad} V1f \in ((ty\_2Elist\_2Elist~A\_27a)^{{A\_27b}}) .  \\
((ap~(ap~(\mname{c\_2Elist\_2ELIST\_\_BIND}~A\_27a~A\_27b)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27b)~V0x)~(c\_2Elist\_2ENIL~A\_27b)))~V1f)=(ap~V1f~V0x))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfedfjeoehemepfpfmejedfefpfpfbeafafmejfpfpfme}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfemgjgdhehpfcdfedfjeoehemepfpfbeafafmejfpfpfnebeaf}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in (A\_27a^{{A\_27b}}) .  ({\unicdcdadad} V1l \in (ty\_2Elist\_2Elist~A\_27b) .  \\
((ap~(ap~(\mname{c\_2Elist\_2ELIST\_\_APPLY}~A\_27a~A\_27b)~(ap~(ap~(c\_2Elist\_2ECONS~\\
(A\_27a^{{A\_27b}}))~V0f)~(c\_2Elist\_2ENIL~(A\_27a^{{A\_27b}}))))~V1l)=(ap~\\
(ap~(c\_2Elist\_2EMAP~A\_27b~A\_27a)~V0f)~V1l))))\end{array}$$
\end{theorem}
\end{document}
