\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=}
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\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
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\newcommand\uniddcgcg{\lambda}
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\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2EringNorm\_2Ecanonical\_\_sum\_\_prod\_\_def (TMT1uwf7a6Bho31o9p9NZoFg78pyKnX4REf)}
\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 $ 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\_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}
\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: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\_2Ecanonical\_2Ecanonical\_\_sum\_\_prod : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Ecanonical\_\_sum\_\_prod~\\
A\_27a \in ((((ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfedgbgogpgogjgdgbgmgpfpfdhfhngpfpfahchpgeg}
\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 $ 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}
\begin{definition}\label{def:dgpfcdfechjgoghgoepgchngpfcdfechpfpfdgbgogpgogjgdgbgmgpfpfdhfhngpfpfahchpgeg}
We define $\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_prod}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ecanonical\_2Ecanonical\_\_sum\_\_prod}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
Let $ c\_2Ecanonical\_2Ecanonical\_\_sum\_\_scalar3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Ecanonical\_\_sum\_\_scalar3~\\
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:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfedgbgogpgogjgdgbgmgpfpfdhfhngpfpfdhdgbgmgbgchdd}
\end{equation}
\begin{definition}\label{def:dgpfcdfechjgoghgoepgchngpfcdfechpfpfdgbgogpgogjgdgbgmgpfpfdhfhngpfpfdhdgbgmgbgchdd}
We define $\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_scalar3}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ecanonical\_2Ecanonical\_\_sum\_\_scalar3}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
Let $ c\_2Ecanonical\_2Ecanonical\_\_sum\_\_scalar2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Ecanonical\_\_sum\_\_scalar2~\\
A\_27a \in ((((ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Elist\_2Elist}~{ty\_2Equote\_2Eindex}})}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfedgbgogpgogjgdgbgmgpfpfdhfhngpfpfdhdgbgmgbgchcd}
\end{equation}
\begin{definition}\label{def:dgpfcdfechjgoghgoepgchngpfcdfechpfpfdgbgogpgogjgdgbgmgpfpfdhfhngpfpfdhdgbgmgbgchcd}
We define $\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_scalar2}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ecanonical\_2Ecanonical\_\_sum\_\_scalar2}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
Let $ c\_2Ecanonical\_2Ecanonical\_\_sum\_\_merge : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ecanonical\_2Ecanonical\_\_sum\_\_merge~\\
A\_27a \in ((((ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a)^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Ecanonical\_2Ecanonical\_\_sum}~{A\_27a}})}})^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedgbgogpgogjgdgbgmgpfcdfedgbgogpgogjgdgbgmgpfpfdhfhngpfpfngfgchhgfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechjgoghgoepgchngpfcdfechpfpfdgbgogpgogjgdgbgmgpfpfdhfhngpfpfngfgchhgfg}
We define $\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_merge}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{A\_27a}})} .  {({{{ap}~{({{c\_2Ecanonical\_2Ecanonical\_\_sum\_\_merge}~{A\_27a}})}}~{({{{ap}~{({{\mname{c\_2Ering\_2Esemi\_\_ring\_\_of}}~{A\_27a}})}}~{V0r}})}})}}}$.
\end{definition}
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} V0sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V1c1 \in A\_27a .  ({\unicdcdadad} V2l1 \in (ty\_2Elist\_2Elist~\\
ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V3t1 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A\_27a) .  ({\unicdcdadad} V4s2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a) .  \\
((ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_prod~A\_27a)~V0sr)~\\
(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V1c1)~V2l1)~V3t1))~\\
V4s2)=(ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_merge~A\_27a)~\\
V0sr)~(ap~(ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_scalar3~\\
A\_27a)~V0sr)~V1c1)~V2l1)~V4s2))~(ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_prod~\\
A\_27a)~V0sr)~V3t1)~V4s2)))))))){\unicdcdcdhd}(({\unicdcdadad} V5sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V6l1 \in (ty\_2Elist\_2Elist~ty\_2Equote\_2Eindex) .  \\
({\unicdcdadad} V7t1 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a) .  ({\unicdcdadad} V8s2 \in \\
(ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a) .  ((ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_prod~\\
A\_27a)~V5sr)~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_varlist~A\_27a)~V6l1)~\\
V7t1))~V8s2)=(ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_merge~\\
A\_27a)~V5sr)~(ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_scalar2~\\
A\_27a)~V5sr)~V6l1)~V8s2))~(ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_prod~\\
A\_27a)~V5sr)~V7t1)~V8s2))))))){\unicdcdcdhd}({\unicdcdadad} V9sr \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V10s2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a) .  \\
((ap~(ap~(ap~(c\_2Ecanonical\_2Ecanonical\_\_sum\_\_prod~A\_27a)~V9sr)~\\
(c\_2Ecanonical\_2ENil\_\_monom~A\_27a))~V10s2)=(c\_2Ecanonical\_2ENil\_\_monom~\\
A\_27a))))))\end{array}
\label{hyp:ehigngpfcdfedgbgogpgogjgdgbgmgpfcdfedgbgogpgogjgdgbgmgpfpfdhfhngpfpfahchpgegpfpfegfggg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechjgoghgoepgchngpfcdfedgbgogpgogjgdgbgmgpfpfdhfhngpfpfahchpgegpfpfegfggg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (ty\_2Ering\_2Ering~\\
A\_27a) .  (({\unicdcdadad} V1c1 \in A\_27a .  ({\unicdcdadad} V2l1 \in (ty\_2Elist\_2Elist~\\
ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V3t1 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A\_27a) .  ({\unicdcdadad} V4s2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a) .  \\
((ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_prod}~A\_27a)~\\
V0r)~(ap~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_monom~A\_27a)~V1c1)~V2l1)~\\
V3t1))~V4s2)=(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_merge}~\\
A\_27a)~V0r)~(ap~(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_scalar3}~\\
A\_27a)~V0r)~V1c1)~V2l1)~V4s2))~(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_prod}~\\
A\_27a)~V0r)~V3t1)~V4s2))))))){\unicdcdcdhd}(({\unicdcdadad} V5l1 \in (ty\_2Elist\_2Elist~\\
ty\_2Equote\_2Eindex) .  ({\unicdcdadad} V6t1 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A\_27a) .  ({\unicdcdadad} V7s2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~A\_27a) .  \\
((ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_prod}~A\_27a)~\\
V0r)~(ap~(ap~(c\_2Ecanonical\_2ECons\_\_varlist~A\_27a)~V5l1)~V6t1))~\\
V7s2)=(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_merge}~\\
A\_27a)~V0r)~(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_scalar2}~\\
A\_27a)~V0r)~V5l1)~V7s2))~(ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_prod}~\\
A\_27a)~V0r)~V6t1)~V7s2)))))){\unicdcdcdhd}({\unicdcdadad} V8s2 \in (ty\_2Ecanonical\_2Ecanonical\_\_sum~\\
A\_27a) .  ((ap~(ap~(ap~(\mname{c\_2EringNorm\_2Er\_\_canonical\_\_sum\_\_prod}~\\
A\_27a)~V0r)~(c\_2Ecanonical\_2ENil\_\_monom~A\_27a))~V8s2)=(c\_2Ecanonical\_2ENil\_\_monom~\\
A\_27a))))))\end{array}$$
\end{theorem}
\end{document}
