\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\_2EringNorm\_2Emonom\_\_insert\_\_def (TMYksGfYnZKfJDwccFBhhhycSYrWCwyo8QF)}
\maketitle
Let $ ty\_2Ecanonical\_2Ecanonical\_\_sum : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}nonempty~(ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A0)\end{array}
\label{hyp:ogfgpfehjhpfcdfedgbgogpgogjgdgbgmgpfcdfedgbgogpgogjgdgbgmgpfpfdhfhng}
\end{equation}
Let $ c\_2Ecanonical\_2ENil\_\_monom : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2ENil\_\_monom~A\_27a \in \\
(ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfeoejgmgpfpfngpgogpgng}
\end{equation}
Let $ ty\_2Equote\_2Eindex : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Equote\_2Eindex}}\label{hyp:ogfgpfehjhpfcdfebhfhpgehfgpfcdfejgogegfgih}
\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\_2Ecanonical\_2ECons\_\_varlist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2ECons\_\_varlist~\\
A\_27a \in (((ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Elist\_2Elist}~{ty\_2Equote\_2Eindex}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfedepgogdhpfpfghbgchmgjgdheh}
\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\_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}
Let $ c\_2EternaryComparisons\_2Elist\_\_compare : {{\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\_2EternaryComparisons\_2Elist\_\_compare~\\
A\_27a~A\_27b \in (((ty\_2EternaryComparisons\_2Eordering^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{({{ty\_2EternaryComparisons\_2Eordering}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfemgjgdhehpfpfdgpgngahbgchfg}
\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\_2EternaryComparisons\_2Eordering2num : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EternaryComparisons\_2Eordering2num \in (ty\_2Enum\_2Enum^{{ty\_2EternaryComparisons\_2Eordering}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghgcdogfhng}
\end{equation}
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: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:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
\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:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\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}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefbd}
We define $\mname{c\_2Earithmetic\_2EBIT1}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\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: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: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: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}
\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}
\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}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeehfgchogbgchjhdepgngahbgchjgdhpgogdhpfcdfepgchegfgchjgoghgpfpfdebedffe}
We define $\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {ty\_2EternaryComparisons\_2Eordering} .  {{\uniddcgcg} V1v0 \in {A\_27a} .  {{\uniddcgcg} V2v1 \in {A\_27a} .  {{\uniddcgcg} V3v2 \in {A\_27a} .  {({{{ap}~{({{\uniddcgcg} V4m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V4m}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{V1v0}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V4m}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{V2v1}})}}~{V3v2}})}})}})}}~{({{{ap}~{c\_2EternaryComparisons\_2Eordering2num}}~{V0x}})}})}}}}}}$.
\end{definition}
Let $ c\_2Ecanonical\_2ECons\_\_monom : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2ECons\_\_monom~A\_27a \in \\
((((ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Elist\_2Elist}~{ty\_2Equote\_2Eindex}})}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfedepgogdhpfpfngpgogpgng}
\end{equation}
Let $ ty\_2Ering\_2Ering : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ering\_2Ering}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfechjgoghgpfcdfechjgoghg}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RM : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RM~A\_27a \in (((\\
A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfne}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RP : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RP~A\_27a \in (((\\
A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfaf}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_R1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_R1~A\_27a \in (A\_27a^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfbd}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_R0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_R0~A\_27a \in (A\_27a^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfad}
\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\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a \in (((((ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a)^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}})^{{A\_27a}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfechfgdgpgchegehjhahfgpfcdfedhfgngjgpfpfchjgoghg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfpggg}
We define $\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring}~{A\_27a}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_R0}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_R1}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
Let $ c\_2Ecanonical\_2Emonom\_\_insert : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Emonom\_\_insert~\\
A\_27a \in (((((ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Elist\_2Elist}~{ty\_2Equote\_2Eindex}})}})^{{A\_27a}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfengpgogpgngpfpfjgogdhfgcheh}
\end{equation}
\begin{definition}\label{def:dgpfcdfechjgoghgoepgchngpfcdfechpfpfngpgogpgngpfpfjgogdhfgcheh}
We define $\mname{c\_2EringNorm\_2Er\_\_monom\_\_insert}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ecanonical\_2Emonom\_\_insert}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
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\_\_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\_\_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}
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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0t2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A\_27a) .  ({\unicdcdadad} V1sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  \\
({\unicdcdadad} V2l2 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V3l1 \in \\
(ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V4c2 \in A\_27a .  \\
({\unicdcdadad} V5c1 \in A\_27a .  ((ap~(ap~(ap~(ap~(c\_2Ecanonical\_2Emonom\_\_insert~\\
A\_27a)~V1sr)~V5c1)~V3l1)~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~\\
A\_27a)~V4c2)~V2l2)~V0t2))=(ap~(ap~(ap~(ap~(\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}~\\
(ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a))~(ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_compare~\\
ty\_2Equote\_2Eindex~ty\_2Equote\_2Eindex)~c\_2Equote\_2Eindex\_\_compare)~\\
V3l1)~V2l2))~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V5c1)~\\
V3l1)~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V4c2)~V2l2)~\\
V0t2)))~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~(ap~(ap~\\
(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~V1sr)~V5c1)~V4c2))~\\
V3l1)~V0t2))~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V4c2)~\\
V2l2)~(ap~(ap~(ap~(ap~(c\_2Ecanonical\_2Emonom\_\_insert~A\_27a)~V1sr)~\\
V5c1)~V3l1)~V0t2)))))))))){\unicdcdcdhd}(({\unicdcdadad} V6t2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A\_27a) .  ({\unicdcdadad} V7sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  \\
({\unicdcdadad} V8l2 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V9l1 \in \\
(ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V10c1 \in A\_27a .  \\
((ap~(ap~(ap~(ap~(c\_2Ecanonical\_2Emonom\_\_insert~A\_27a)~V7sr)~V10c1)~\\
V9l1)~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_varlist~A\_27a)~V8l2)~V6t2))=\\
(ap~(ap~(ap~(ap~(\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}~(ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A\_27a))~(ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_compare~\\
ty\_2Equote\_2Eindex~ty\_2Equote\_2Eindex)~c\_2Equote\_2Eindex\_\_compare)~\\
V9l1)~V8l2))~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V10c1)~\\
V9l1)~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_varlist~A\_27a)~V8l2)~V6t2)))~\\
(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~\\
A\_27a)~V7sr)~V10c1)~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~\\
V7sr)))~V9l1)~V6t2))~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_varlist~A\_27a)~\\
V8l2)~(ap~(ap~(ap~(ap~(c\_2Ecanonical\_2Emonom\_\_insert~A\_27a)~V7sr)~\\
V10c1)~V9l1)~V6t2))))))))){\unicdcdcdhd}({\unicdcdadad} V11sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V12l1 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  \\
({\unicdcdadad} V13c1 \in A\_27a .  ((ap~(ap~(ap~(ap~(c\_2Ecanonical\_2Emonom\_\_insert~\\
A\_27a)~V11sr)~V13c1)~V12l1)~(c\_2Ecanonical\_2ENil\_\_monom~A\_27a))=\\
(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V13c1)~V12l1)~\\
(c\_2Ecanonical\_2ENil\_\_monom~A\_27a))))))))\end{array}
\label{hyp:ehigngpfcdfedgbgogpgogjgdgbgmgpfcdfengpgogpgngpfpfjgogdhfgchehpfpfegfggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdadad} V1a0 \in \\
A\_27a .  ({\unicdcdadad} V2f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V3f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~A\_27a)~(ap~(ap~(ap~(ap~\\
(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~A\_27a)~V0a)~V1a0)~\\
V2f)~V3f0))=V0a))))){\unicdcdcdhd}(({\unicdcdadad} V4a \in A\_27a .  ({\unicdcdadad} V5a0 \in A\_27a .  \\
({\unicdcdadad} V6f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V7f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~(ap~(ap~(ap~(ap~\\
(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~A\_27a)~V4a)~V5a0)~\\
V6f)~V7f0))=V5a0))))){\unicdcdcdhd}(({\unicdcdadad} V8a \in A\_27a .  ({\unicdcdadad} V9a0 \in A\_27a .  \\
({\unicdcdadad} V10f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V11f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~(ap~(ap~(ap~(ap~\\
(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~A\_27a)~V8a)~V9a0)~\\
V10f)~V11f0))=V10f))))){\unicdcdcdhd}({\unicdcdadad} V12a \in A\_27a .  ({\unicdcdadad} V13a0 \in \\
A\_27a .  ({\unicdcdadad} V14f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V15f0 \in ((\\
A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~\\
A\_27a)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V12a)~V13a0)~V14f)~V15f0))=V15f0))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfbgdgdgfgdhdhpgchdh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechjgoghgoepgchngpfcdfengpgogpgngpfpfjgogdhfgchehpfpfegfggg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (ty\_2Ering\_2Ering~\\
A\_27a) .  (({\unicdcdadad} V1t2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a) .  \\
({\unicdcdadad} V2l2 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V3l1 \in \\
(ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V4c2 \in A\_27a .  \\
({\unicdcdadad} V5c1 \in A\_27a .  ((ap~(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_monom\_\_insert}~\\
A\_27a)~V0r)~V5c1)~V3l1)~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~\\
A\_27a)~V4c2)~V2l2)~V1t2))=(ap~(ap~(ap~(ap~(\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}~\\
(ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a))~(ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_compare~\\
ty\_2Equote\_2Eindex~ty\_2Equote\_2Eindex)~c\_2Equote\_2Eindex\_\_compare)~\\
V3l1)~V2l2))~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V5c1)~\\
V3l1)~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V4c2)~V2l2)~\\
V1t2)))~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~(ap~(ap~\\
(ap~(c\_2Ering\_2Ering\_\_RP~A\_27a)~V0r)~V5c1)~V4c2))~V3l1)~V1t2))~\\
(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V4c2)~V2l2)~(ap~\\
(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_monom\_\_insert}~A\_27a)~V0r)~V5c1)~\\
V3l1)~V1t2))))))))){\unicdcdcdhd}(({\unicdcdadad} V6t2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A\_27a) .  ({\unicdcdadad} V7l2 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  \\
({\unicdcdadad} V8l1 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V9c1 \in \\
A\_27a .  ((ap~(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_monom\_\_insert}~A\_27a)~\\
V0r)~V9c1)~V8l1)~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_varlist~A\_27a)~\\
V7l2)~V6t2))=(ap~(ap~(ap~(ap~(\mname{c\_2EternaryComparisons\_2Eordering\_\_CASE}~\\
(ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a))~(ap~(ap~(ap~(c\_2EternaryComparisons\_2Elist\_\_compare~\\
ty\_2Equote\_2Eindex~ty\_2Equote\_2Eindex)~c\_2Equote\_2Eindex\_\_compare)~\\
V8l1)~V7l2))~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V9c1)~\\
V8l1)~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_varlist~A\_27a)~V7l2)~V6t2)))~\\
(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~(ap~(ap~(ap~(c\_2Ering\_2Ering\_\_RP~\\
A\_27a)~V0r)~V9c1)~(ap~(c\_2Ering\_2Ering\_\_R1~A\_27a)~V0r)))~V8l1)~\\
V6t2))~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_varlist~A\_27a)~V7l2)~(ap~\\
(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_monom\_\_insert}~A\_27a)~V0r)~V9c1)~\\
V8l1)~V6t2)))))))){\unicdcdcdhd}({\unicdcdadad} V10l1 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  \\
({\unicdcdadad} V11c1 \in A\_27a .  ((ap~(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_monom\_\_insert}~\\
A\_27a)~V0r)~V11c1)~V10l1)~(c\_2Ecanonical\_2ENil\_\_monom~A\_27a))=\\
(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V11c1)~V10l1)~\\
(c\_2Ecanonical\_2ENil\_\_monom~A\_27a))))))))\end{array}$$
\end{theorem}
\end{document}
