\documentclass{article}
\newtheorem{axiom}{Axiom}
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\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\newcommand\unicdcdbdgd{\setminus}
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\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\_2EOmega\_2Esumc\_\_MULT (TMQNeSDDTSoboe2pqXtXYHL2sY6ew4HUprc)}
\maketitle
Let $ ty\_2Einteger\_2Eint : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Einteger\_2Eint}}\label{hyp:ogfgpfehjhpfcdfejgogehfghgfgchpfcdfejgogeh}
\end{equation}
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\_2EOmega\_2Esumc : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EOmega\_2Esumc \in ((ty\_2Einteger\_2Eint^{{({{ty\_2Elist\_2Elist}~{ty\_2Einteger\_2Eint}})}})^{{({{ty\_2Elist\_2Elist}~{ty\_2Einteger\_2Eint}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfepengfghgbgpfcdfedhfhngdg}
\end{equation}
\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}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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\_2Einteger\_2Eint\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeafpfpfdemebedfdf}
\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:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeaf}
We define $\mname{c\_2Einteger\_2Eint\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfbgegeg}
\end{equation}
Let $ c\_2Einteger\_2Etint\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpffgbh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_ABS\_\_CLASS \in (ty\_2Einteger\_2Eint^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedf}
We define $\mname{c\_2Einteger\_2Eint\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})} .  {({{{ap}~{c\_2Einteger\_2Eint\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbgegeg}
We define $\mname{c\_2Einteger\_2Eint\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_add}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\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 $ 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 $ c\_2Einteger\_2Etint\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_mul \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfngfhmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfngfhmg}
We define $\mname{c\_2Einteger\_2Eint\_\_mul}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_mul}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
Let $ c\_2Einteger\_2Eint\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Einteger\_2Eint\_\_of\_\_num} \in {({{ty\_2Einteger\_2Eint}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfpgggpfpfogfhng}
\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: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\_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}
\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}
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}
\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: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:dgpfcdfengbgchlgfgchpfcdfebede}
We define $\mname{c\_2Emarker\_2EAC}$ to be ${{\uniddcgcg} V0b1 \in {2} .  {{\uniddcgcg} V1b2 \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{V0b1}})}}~{V1b2}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0cs \in (ty\_2Elist\_2Elist~ty\_2Einteger\_2Eint) .  ({\unicdcdadad} V1vs \in \\
(ty\_2Elist\_2Elist~ty\_2Einteger\_2Eint) .  ({\unicdcdadad} V2c \in ty\_2Einteger\_2Eint .  \\
({\unicdcdadad} V3v \in ty\_2Einteger\_2Eint .  (((ap~(ap~c\_2EOmega\_2Esumc~\\
(c\_2Elist\_2ENIL~ty\_2Einteger\_2Eint))~V1vs)=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(((ap~(ap~c\_2EOmega\_2Esumc~V0cs)~(c\_2Elist\_2ENIL~\\
ty\_2Einteger\_2Eint))=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0})){\unicdcdcdhd}\\
((ap~(ap~c\_2EOmega\_2Esumc~(ap~(ap~(c\_2Elist\_2ECONS~ty\_2Einteger\_2Eint)~\\
V2c)~V0cs))~(ap~(ap~(c\_2Elist\_2ECONS~ty\_2Einteger\_2Eint)~V3v)~\\
V1vs))=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V2c)~V3v))~(ap~(ap~c\_2EOmega\_2Esumc~V0cs)~V1vs)))))))))\end{array}
\label{hyp:ehigngpfcdfepengfghgbgpfcdfedhfhngdgpfpfehigng}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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} 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} V0y \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1x \in ty\_2Einteger\_2Eint .  \\
((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V1x)~V0y)=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V0y)~V1x))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfneffmepfpfdepenene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0z \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
({\unicdcdadad} V2x \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V2x)~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V1y)~V0z))=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V2x)~V1y))~V0z)))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfneffmepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0z \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
({\unicdcdadad} V2x \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V2x)~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V1y)~V0z))=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V2x)~V1y))~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V2x)~V0z))))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfmeeejedfefcfjece}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
({\unicdcdadad} V2z \in ty\_2Einteger\_2Eint .  (((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
V0x)~V2z)=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V1y)~V2z)){\unicdbdjded}(V0x=V1y)))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpffebfpfpfcfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V0x)~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfneffmepfpfcfkffecfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
({\unicdcdadad} V2z \in ty\_2Einteger\_2Eint .  (((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V0x)~V1y)=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V0x)~V2z)){\unicdbdjded}((V0x=(ap~\\
c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0})){\unicdcdcdid}(V1y=V2z))))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpffebfpfpfmeneffme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~c\_2Einteger\_2Eint\_\_of\_\_num~V0m)=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
V1n)){\unicdbdjded}(V0m=V1n))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfjeoeke}
\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}}) .  ((ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~\\
V0f)~(c\_2Elist\_2ENIL~A\_27a))=(c\_2Elist\_2ENIL~A\_27b))){\unicdcdcdhd}({\unicdcdadad} V1f \in \\
(A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V2h \in A\_27a .  ({\unicdcdadad} V3t \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~V1f)~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V2h)~V3t))=(ap~(ap~(c\_2Elist\_2ECONS~A\_27b)~(ap~V1f~V2h))~\\
(ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~V1f)~V3t)))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenebeaf}
\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} V0l \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ((V0l=(c\_2Elist\_2ENIL~A\_27a)){\unicdcdcdid}({\unicdcdaddd} V1h \in A\_27a .  (\\
{\unicdcdaddd} V2t \in (ty\_2Elist\_2Elist~A\_27a) .  (V0l=(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V1h)~V2t))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemgjgdhehpfpfogdgigpgehpgngjh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfepengfghgbgpfcdfedhfhngdgpfpfneffmeef}
$$\begin{array}{c}({\unicdcdadad} V0cs \in (ty\_2Elist\_2Elist~ty\_2Einteger\_2Eint) .  ({\unicdcdadad} V1vs \in \\
(ty\_2Elist\_2Elist~ty\_2Einteger\_2Eint) .  ({\unicdcdadad} V2f \in ty\_2Einteger\_2Eint .  \\
((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V2f)~(ap~(ap~c\_2EOmega\_2Esumc~\\
V0cs)~V1vs))=(ap~(ap~c\_2EOmega\_2Esumc~(ap~(ap~(c\_2Elist\_2EMAP~\\
ty\_2Einteger\_2Eint~ty\_2Einteger\_2Eint)~({\uniddcgcg} V3x \in ty\_2Einteger\_2Eint .  \\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V2f)~V3x)))~V0cs))~V1vs)))))\end{array}$$
\end{theorem}
\end{document}
