\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\_2Ecanonical\_2Eivl\_\_aux\_\_ok (TMYCxW5miXaQKaURu1UhfkWpSaLewxpbJsD)}
\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\_2Equote\_2Evarmap : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Equote\_2Evarmap}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfebhfhpgehfgpfcdfeghbgchngbgah}
\end{equation}
Let $ ty\_2Equote\_2Eindex : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Equote\_2Eindex}}\label{hyp:ogfgpfehjhpfcdfebhfhpgehfgpfcdfejgogegfgih}
\end{equation}
Let $ c\_2Equote\_2Evarmap\_\_find : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Equote\_2Evarmap\_\_find~A\_27a \in \\
((A\_27a^{{({{ty\_2Equote\_2Evarmap}~{A\_27a}})}})^{{ty\_2Equote\_2Eindex}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebhfhpgehfgpfcdfeghbgchngbgahpfpfggjgogeg}
\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 $ ty\_2Esemi\_\_ring\_2Esemi\_\_ring : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghg}
\end{equation}
Let $ c\_2Ecanonical\_2Eivl\_\_aux : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Eivl\_\_aux~A\_27a \in \\
((((A\_27a^{{({{ty\_2Elist\_2Elist}~{ty\_2Equote\_2Eindex}})}})^{{ty\_2Equote\_2Eindex}})^{{({{ty\_2Equote\_2Evarmap}~{A\_27a}})}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfejgghmgpfpfbgfhih}
\end{equation}
Let $ c\_2Ecanonical\_2Einterp\_\_vl : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Einterp\_\_vl~A\_27a \in \\
(((A\_27a^{{({{ty\_2Elist\_2Elist}~{ty\_2Equote\_2Eindex}})}})^{{({{ty\_2Equote\_2Evarmap}~{A\_27a}})}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfejgogehfgchahpfpfghmg}
\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}
\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: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}
Let $ c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~\\
A\_27a \in (A\_27a^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfbd}
\end{equation}
Let $ c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~\\
A\_27a \in (((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfne}
\end{equation}
Let $ c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~\\
A\_27a \in (((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfaf}
\end{equation}
Let $ c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~\\
A\_27a \in (A\_27a^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfad}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhfgngjgpfpfchjgoghgpfcdfejgdhpfpfdhfgngjgpfpfchjgoghg}
We define $\mname{c\_2Esemi\_\_ring\_2Eis\_\_semi\_\_ring}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1n \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2m \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{V1n}})}}~{V2m}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{V2m}})}}~{V1n}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3n \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4m \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5p \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{V3n}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{V4m}})}}~{V5p}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{V3n}})}}~{V4m}})}})}}~{V5p}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6n \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7m \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{V6n}})}}~{V7m}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{V7m}})}}~{V6n}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8n \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V9m \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V10p \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{V8n}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{V9m}})}}~{V10p}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{V8n}})}}~{V9m}})}})}}~{V10p}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V11n \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0}~{A\_27a}})}}~{V0r}})}})}}~{V11n}})}})}}~{V11n}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V12n \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1}~{A\_27a}})}}~{V0r}})}})}}~{V12n}})}})}}~{V12n}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V13n \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0}~{A\_27a}})}}~{V0r}})}})}}~{V13n}})}})}}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0}~{A\_27a}})}}~{V0r}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V14n \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V15m \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V16p \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{V14n}})}}~{V15m}})}})}}~{V16p}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{V14n}})}}~{V16p}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM}~{A\_27a}})}}~{V0r}})}}~{V15m}})}}~{V16p}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}}}$.
\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} V0sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V1vm \in (ty\_2Equote\_2Evarmap~A\_27a) .  ({\unicdcdadad} V2x \in \\
ty\_2Equote\_2Eindex .  ((ap~(ap~(ap~(ap~(c\_2Ecanonical\_2Eivl\_\_aux~\\
A\_27a)~V0sr)~V1vm)~V2x)~(c\_2Elist\_2ENIL~ty\_2Equote\_2Eindex))=\\
(ap~(ap~(c\_2Equote\_2Evarmap\_\_find~A\_27a)~V2x)~V1vm))))){\unicdcdcdhd}({\unicdcdadad} V3sr \in \\
(ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V4vm \in (ty\_2Equote\_2Evarmap~\\
A\_27a) .  ({\unicdcdadad} V5x \in ty\_2Equote\_2Eindex .  ({\unicdcdadad} V6x\_27 \in ty\_2Equote\_2Eindex .  \\
({\unicdcdadad} V7t\_27 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ((ap~\\
(ap~(ap~(ap~(c\_2Ecanonical\_2Eivl\_\_aux~A\_27a)~V3sr)~V4vm)~V5x)~(\\
ap~(ap~(c\_2Elist\_2ECONS~ty\_2Equote\_2Eindex)~V6x\_27)~V7t\_27))=\\
(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V3sr)~(ap~\\
(ap~(c\_2Equote\_2Evarmap\_\_find~A\_27a)~V5x)~V4vm))~(ap~(ap~(ap~(ap~\\
(c\_2Ecanonical\_2Eivl\_\_aux~A\_27a)~V3sr)~V4vm)~V6x\_27)~V7t\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfedgbgogpgogjgdgbgmgpfcdfejgghmgpfpfbgfhihpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V1vm \in (ty\_2Equote\_2Evarmap~A\_27a) .  ((ap~(ap~(\\
ap~(c\_2Ecanonical\_2Einterp\_\_vl~A\_27a)~V0sr)~V1vm)~(c\_2Elist\_2ENIL~\\
ty\_2Equote\_2Eindex))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~\\
A\_27a)~V0sr)))){\unicdcdcdhd}({\unicdcdadad} V2sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V3vm \in (ty\_2Equote\_2Evarmap~A\_27a) .  ({\unicdcdadad} V4x \in \\
ty\_2Equote\_2Eindex .  ({\unicdcdadad} V5t \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  \\
((ap~(ap~(ap~(c\_2Ecanonical\_2Einterp\_\_vl~A\_27a)~V2sr)~V3vm)~(ap~\\
(ap~(c\_2Elist\_2ECONS~ty\_2Equote\_2Eindex)~V4x)~V5t))=(ap~(ap~(\\
ap~(ap~(c\_2Ecanonical\_2Eivl\_\_aux~A\_27a)~V2sr)~V3vm)~V4x)~V5t)))))))\end{array}
\label{hyp:ehigngpfcdfedgbgogpgogjgdgbgmgpfcdfejgogehfgchahpfpfghmgpfpfegfggg}
\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} V0r \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ((p~(ap~(\mname{c\_2Esemi\_\_ring\_2Eis\_\_semi\_\_ring}~A\_27a)~V0r)){\unicdbdjdcd}\\
({\unicdcdadad} V1n \in A\_27a .  ((ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~\\
A\_27a)~V0r)~V1n)~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~\\
V0r))=V1n))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfengfhmgehpfpfpgogfgpfpfchjghgigeh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedgbgogpgogjgdgbgmgpfcdfejgghmgpfpfbgfhihpfpfpglg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ((p~(ap~(\mname{c\_2Esemi\_\_ring\_2Eis\_\_semi\_\_ring}~A\_27a)~V0sr)){\unicdbdjdcd}\\
({\unicdcdadad} V1vm \in (ty\_2Equote\_2Evarmap~A\_27a) .  ({\unicdcdadad} V2v \in (ty\_2Elist\_2Elist~\\
ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V3i \in ty\_2Equote\_2Eindex .  ((ap~\\
(ap~(ap~(ap~(c\_2Ecanonical\_2Eivl\_\_aux~A\_27a)~V0sr)~V1vm)~V3i)~V2v)=\\
(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0sr)~(ap~\\
(ap~(c\_2Equote\_2Evarmap\_\_find~A\_27a)~V3i)~V1vm))~(ap~(ap~(ap~(c\_2Ecanonical\_2Einterp\_\_vl~\\
A\_27a)~V0sr)~V1vm)~V2v))))))))\end{array}$$
\end{theorem}
\end{document}
