\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\_2Evarlist\_\_merge\_\_ok (TMPoiKCuBoKf3DM63ithXbuYGSCEhJYNE1D)}
\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: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: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: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}
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 $ ty\_2EternaryComparisons\_2Eordering : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ogfgpfehjhpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghg}
\end{equation}
Let $ c\_2EternaryComparisons\_2ELESS : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2EternaryComparisons\_2ELESS} \in {ty\_2EternaryComparisons\_2Eordering}}\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfemefedfdf}
\end{equation}
Let $ c\_2Equote\_2Eindex\_\_compare : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Equote\_2Eindex\_\_compare \in ((ty\_2EternaryComparisons\_2Eordering^{{ty\_2Equote\_2Eindex}})^{{ty\_2Equote\_2Eindex}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebhfhpgehfgpfcdfejgogegfgihpfpfdgpgngahbgchfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfebhfhpgehfgpfcdfejgogegfgihpfpfmgeh}
We define $\mname{c\_2Equote\_2Eindex\_\_lt}$ to be ${{\uniddcgcg} V0i1 \in {ty\_2Equote\_2Eindex} .  {{\uniddcgcg} V1i2 \in {ty\_2Equote\_2Eindex} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2EternaryComparisons\_2Eordering}})}}~{({{{ap}~{({{{ap}~{c\_2Equote\_2Eindex\_\_compare}}~{V0i1}})}}~{V1i2}})}})}}~{c\_2EternaryComparisons\_2ELESS}})}}}$.
\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: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: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}
\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}
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\_2EternaryComparisons\_2Elist\_\_merge : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2EternaryComparisons\_2Elist\_\_merge~\\
A\_27a \in ((((ty\_2Elist\_2Elist~A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfemgjgdhehpfpfngfgchhgfg}
\end{equation}
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}
{({{\unicdcdadad} V0t \in {2} .  {({{({{p}~{V0t}})}{\unicdcdcdid}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifdemeffeefeeepfpfnejeeeeemefe}
\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}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  (((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1){\unicdcdcdhd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~\\
V0t1)~V1t2)=V1t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in 2 .  ({\unicdcdadad} V1Q \in 2 .  \\
({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3x\_27 \in A\_27a .  ({\unicdcdadad} V4y \in A\_27a .  \\
({\unicdcdadad} V5y\_27 \in A\_27a .  ((((p~V0P){\unicdbdjded}(p~V1Q)){\unicdcdcdhd}(((p~V1Q){\unicdbdjdcd}(V2x=V3x\_27)){\unicdcdcdhd}\\
((\preoh (p~V1Q)){\unicdbdjdcd}(V4y=V5y\_27)))){\unicdbdjdcd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\\
V0P)~V2x)~V4y)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~V1Q)~V3x\_27)~\\
V5y\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdepeoehe}
\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 .  ({\unicdcdadad} V2m \in A\_27a .  ({\unicdcdadad} V3p \in A\_27a .  (\\
(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0r)~V1n)~\\
(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0r)~V2m)~\\
V3p))=(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0r)~\\
(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0r)~V1n)~\\
V2m))~V3p)))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfengfhmgehpfpfbgdhdhpgdg}
\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)~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~V0r))~\\
V1n)=V1n))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfengfhmgehpfpfpgogfgpfpfmgfgggeh}
\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}
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} V1m \in A\_27a .  ({\unicdcdadad} V2n \in A\_27a .  ({\unicdcdadad} V3p \in A\_27a .  (\\
(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0r)~(ap~\\
(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0r)~V1m)~V2n))~\\
V3p)=(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0r)~\\
(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0r)~V2n)~\\
V3p))~V1m)))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfengfhmgehpfpfchpgehbgehfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0l1 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V1a\_\_lt \in ((2^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_merge~\\
A\_27a)~V1a\_\_lt)~V0l1)~(c\_2Elist\_2ENIL~A\_27a))=V0l1))){\unicdcdcdhd}(({\unicdcdadad} V2v5 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V3v4 \in A\_27a .  ({\unicdcdadad} V4a\_\_lt \in \\
((2^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_merge~\\
A\_27a)~V4a\_\_lt)~(c\_2Elist\_2ENIL~A\_27a))~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V3v4)~V2v5))=(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V3v4)~V2v5))))){\unicdcdcdhd}\\
({\unicdcdadad} V5y \in A\_27a .  ({\unicdcdadad} V6x \in A\_27a .  ({\unicdcdadad} V7l2 \in (ty\_2Elist\_2Elist~\\
A\_27a) .  ({\unicdcdadad} V8l1 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V9a\_\_lt \in \\
((2^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_merge~\\
A\_27a)~V9a\_\_lt)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V6x)~V8l1))~(ap~\\
(ap~(c\_2Elist\_2ECONS~A\_27a)~V5y)~V7l2))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~\\
(ty\_2Elist\_2Elist~A\_27a))~(ap~(ap~V9a\_\_lt~V6x)~V5y))~(ap~(ap~(c\_2Elist\_2ECONS~\\
A\_27a)~V6x)~(ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_merge~\\
A\_27a)~V9a\_\_lt)~V8l1)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V5y)~V7l2))))~\\
(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V5y)~(ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_merge~\\
A\_27a)~V9a\_\_lt)~(ap~(ap~(c\_2Elist\_2ECONS~A\_27a)~V6x)~V8l1))~V7l2)))))))))))\end{array}
\label{hyp:ehigngpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfemgjgdhehpfpfngfgchhgfgpfpfegfggg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedgbgogpgogjgdgbgmgpfcdfeghbgchmgjgdhehpfpfngfgchhgfgpfpfpglg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ((p~(ap~(\mname{c\_2Esemi\_\_ring\_2Eis\_\_semi\_\_ring}~A\_27a)~V0sr)){\unicdbdjdcd}\\
({\unicdcdadad} V1vm \in (ty\_2Equote\_2Evarmap~A\_27a) .  ({\unicdcdadad} V2x \in (ty\_2Elist\_2Elist~\\
ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V3y \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  \\
((ap~(ap~(ap~(c\_2Ecanonical\_2Einterp\_\_vl~A\_27a)~V0sr)~V1vm)~(ap~\\
(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_merge~ty\_2Equote\_2Eindex)~\\
\mname{c\_2Equote\_2Eindex\_\_lt})~V2x)~V3y))=(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~\\
A\_27a)~V0sr)~(ap~(ap~(ap~(c\_2Ecanonical\_2Einterp\_\_vl~A\_27a)~V0sr)~\\
V1vm)~V2x))~(ap~(ap~(ap~(c\_2Ecanonical\_2Einterp\_\_vl~A\_27a)~V0sr)~\\
V1vm)~V3y))))))))\end{array}$$
\end{theorem}
\end{document}
