\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\_2ESUM\_\_ACC\_\_SUM\_\_LEM (TMTzforYTiWN2WFtvoNxQQ8EB17v98g7Egx)}
\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 $ 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}
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\_2ESUM : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Elist\_2ESUM} \in {({{ty\_2Enum\_2Enum}^{{({{ty\_2Elist\_2Elist}~{ty\_2Enum\_2Enum}})}}})}}\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedfffne}
\end{equation}
\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}
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}
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\_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\_2ESUM\_\_ACC : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Elist\_2ESUM\_\_ACC \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{({{ty\_2Elist\_2Elist}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfedfffnepfpfbedede}
\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: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}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~\\
\mname{c\_2Enum\_2E0})=V0m))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n)=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1n)~V0m))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfdfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n)=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1n)~V0m))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfdepenene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V1n)~V2p))=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))~V2p)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  (((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~\\
V2p)=(ap~(ap~c\_2Earithmetic\_2E\_2B~V1n)~V2p)){\unicdbdjded}(V0m=V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfefebfpfpfnepeoepepfpfbeeeeepfpffebf}
\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} 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}(((ap~c\_2Elist\_2ESUM~(c\_2Elist\_2ENIL~ty\_2Enum\_2Enum))=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}\\
({\unicdcdadad} V0h \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1t \in (ty\_2Elist\_2Elist~\\
ty\_2Enum\_2Enum) .  ((ap~c\_2Elist\_2ESUM~(ap~(ap~(c\_2Elist\_2ECONS~\\
ty\_2Enum\_2Enum)~V0h)~V1t))=(ap~(ap~c\_2Earithmetic\_2E\_2B~V0h)~\\
(ap~c\_2Elist\_2ESUM~V1t))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfedfffne}
\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:ehigngpfcdfemgjgdhehpfcdfemgjgdhehpfpfjeoeeeffdeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0acc \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Elist\_2ESUM\_\_ACC~\\
(c\_2Elist\_2ENIL~ty\_2Enum\_2Enum))~V0acc)=V0acc)){\unicdcdcdhd}({\unicdcdadad} V1h \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V2t \in (ty\_2Elist\_2Elist~ty\_2Enum\_2Enum) .  \\
({\unicdcdadad} V3acc \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Elist\_2ESUM\_\_ACC~\\
(ap~(ap~(c\_2Elist\_2ECONS~ty\_2Enum\_2Enum)~V1h)~V2t))~V3acc)=(ap~\\
(ap~c\_2Elist\_2ESUM\_\_ACC~V2t)~(ap~(ap~c\_2Earithmetic\_2E\_2B~V1h)~\\
V3acc)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfedfffnepfpfbededepfpfeefege}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfemgjgdhehpfcdfedfffnepfpfbededepfpfdfffnepfpfmefene}
$$\begin{array}{c}({\unicdcdadad} V0L \in (ty\_2Elist\_2Elist~ty\_2Enum\_2Enum) .  ({\unicdcdadad} V1n \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Elist\_2ESUM\_\_ACC~V0L)~V1n)=(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~(ap~c\_2Elist\_2ESUM~V0L))~V1n))))\end{array}$$
\end{theorem}
\end{document}
