\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\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_literal\_\_11 (TMQLqEzuFLRuEXsLBfnx4317ff47Sv3KRbz)}
\maketitle
Let $ c\_2Ebool\_2EARB : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{{{c\_2Ebool\_2EARB}~{A\_27a}} \in {A\_27a}}}}\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfebecfce}
\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:dgpfcdfedgpgngcgjgogpfcdfele}
We define $\mname{c\_2Ecombin\_2EK}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1y \in {A\_27b} .  {V0x}})}})}}}$.
\end{definition}
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\_2Esemi\_\_ring\_\_SR0\_\_fupd : {{\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\_\_fupd~\\
A\_27a \in (((ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a)^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})^{{({{A\_27a}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfadpfpfggfhaheg}
\end{equation}
Let $ c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd : {{\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\_\_fupd~\\
A\_27a \in (((ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a)^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})^{{({{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfnepfpfggfhaheg}
\end{equation}
Let $ c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd : {{\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\_\_fupd~\\
A\_27a \in (((ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a)^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})^{{({{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfafpfpfggfhaheg}
\end{equation}
Let $ c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd : {{\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\_\_fupd~\\
A\_27a \in (((ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a)^{{({{ty\_2Esemi\_\_ring\_2Esemi\_\_ring}~{A\_27a}})}})^{{({{A\_27a}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfbdpfpfggfhaheg}
\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\_\_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}
\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: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}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27a~A\_27b)~V0x)~V1y)=V0x)))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfelepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V1f \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~\\
A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd~A\_27a)~\\
V1f)~V0s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~A\_27a)~V0s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V2s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V3f \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~\\
A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd~A\_27a)~\\
V3f)~V2s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~A\_27a)~V2s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V4s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V5f \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~\\
A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd~A\_27a)~\\
V5f)~V4s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~A\_27a)~V4s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V6s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V7f \in \\
(A\_27a^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~\\
(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0\_\_fupd~A\_27a)~V7f)~\\
V6s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~V6s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V8s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V9f \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~\\
A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd~A\_27a)~\\
V9f)~V8s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~V8s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V10s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V11f \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~\\
A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd~A\_27a)~\\
V11f)~V10s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~V10s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V12s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V13f \in \\
(A\_27a^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~\\
(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0\_\_fupd~A\_27a)~V13f)~\\
V12s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~V12s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V14s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V15f \in \\
(A\_27a^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~\\
(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd~A\_27a)~V15f)~\\
V14s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~V14s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V16s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V17f \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~\\
A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd~A\_27a)~\\
V17f)~V16s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~V16s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V18s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V19f \in \\
(A\_27a^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~\\
(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0\_\_fupd~A\_27a)~V19f)~\\
V18s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V18s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V20s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V21f \in \\
(A\_27a^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~\\
(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd~A\_27a)~V21f)~\\
V20s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V20s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V22s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V23f \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~\\
A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd~A\_27a)~\\
V23f)~V22s))=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V22s)))){\unicdcdcdhd}\\
(({\unicdcdadad} V24s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  ({\unicdcdadad} V25f \in \\
(A\_27a^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~A\_27a)~\\
(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0\_\_fupd~A\_27a)~V25f)~\\
V24s))=(ap~V25f~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~A\_27a)~\\
V24s))))){\unicdcdcdhd}(({\unicdcdadad} V26s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  \\
({\unicdcdadad} V27f \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~\\
A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd~A\_27a)~\\
V27f)~V26s))=(ap~V27f~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~\\
A\_27a)~V26s))))){\unicdcdcdhd}(({\unicdcdadad} V28s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V29f \in (((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  \\
((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd~\\
A\_27a)~V29f)~V28s))=(ap~V29f~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~\\
A\_27a)~V28s))))){\unicdcdcdhd}({\unicdcdadad} V30s \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V31f \in (((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  \\
((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd~\\
A\_27a)~V31f)~V30s))=(ap~V31f~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~\\
A\_27a)~V30s))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfbgdgdgggfhahegdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s1 \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~\\
A\_27a) .  ({\unicdcdadad} V1s2 \in (ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a) .  \\
((V0s1=V1s2){\unicdbdjded}(((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~A\_27a)~\\
V0s1)=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0~A\_27a)~V1s2)){\unicdcdcdhd}\\
(((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~A\_27a)~V0s1)=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1~\\
A\_27a)~V1s2)){\unicdcdcdhd}(((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~\\
V0s1)=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP~A\_27a)~V1s2)){\unicdcdcdhd}\\
((ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~A\_27a)~V0s1)=(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM~\\
A\_27a)~V1s2))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdgpgngahpgogfgogehpfpffgbhfhbgmgjgehjh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfmgjgehfgchbgmgpfpfbdbd}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a01 \in A\_27a .  ({\unicdcdadad} V1a1 \in \\
A\_27a .  ({\unicdcdadad} V2f01 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V3f1 \in ((\\
A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V4a02 \in A\_27a .  ({\unicdcdadad} V5a2 \in A\_27a .  \\
({\unicdcdadad} V6f02 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V7f2 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
(((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0\_\_fupd~A\_27a)~(ap~\\
(\mname{c\_2Ecombin\_2EK}~A\_27a~A\_27a)~V0a01))~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd~\\
A\_27a)~(ap~(\mname{c\_2Ecombin\_2EK}~A\_27a~A\_27a)~V1a1))~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd~\\
A\_27a)~(ap~(\mname{c\_2Ecombin\_2EK}~((A\_27a^{{A\_27a}})^{{A\_27a}})~((A\_27a^{{A\_27a}})^{{A\_27a}}))~\\
V2f01))~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd~A\_27a)~\\
(ap~(\mname{c\_2Ecombin\_2EK}~((A\_27a^{{A\_27a}})^{{A\_27a}})~((A\_27a^{{A\_27a}})^{{A\_27a}}))~\\
V3f1))~(c\_2Ebool\_2EARB~(ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a))))))=\\
(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0\_\_fupd~A\_27a)~(ap~(\\
\mname{c\_2Ecombin\_2EK}~A\_27a~A\_27a)~V4a02))~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd~\\
A\_27a)~(ap~(\mname{c\_2Ecombin\_2EK}~A\_27a~A\_27a)~V5a2))~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd~\\
A\_27a)~(ap~(\mname{c\_2Ecombin\_2EK}~((A\_27a^{{A\_27a}})^{{A\_27a}})~((A\_27a^{{A\_27a}})^{{A\_27a}}))~\\
V6f02))~(ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd~A\_27a)~\\
(ap~(\mname{c\_2Ecombin\_2EK}~((A\_27a^{{A\_27a}})^{{A\_27a}})~((A\_27a^{{A\_27a}})^{{A\_27a}}))~\\
V7f2))~(c\_2Ebool\_2EARB~(ty\_2Esemi\_\_ring\_2Esemi\_\_ring~A\_27a))))))){\unicdbdjded}\\
((V0a01=V4a02){\unicdcdcdhd}((V1a1=V5a2){\unicdcdcdhd}((V2f01=V6f02){\unicdcdcdhd}(V3f1=V7f2)))))))))))))\end{array}$$
\end{theorem}
\end{document}
