\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\_\_fn\_\_updates (TMLokCxLVHDRya5XF6nnjwPCCcR7kUXdKQD)}
\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: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 $ 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\_\_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\_\_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\_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}
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}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f1 \in (A\_27a^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0\_\_fupd~\\
A\_27a)~V0f1)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V1a)~V2a0)~V3f)~V4f0))=(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~(ap~V0f1~V1a))~V2a0)~V3f)~V4f0)))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfadpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f1 \in (A\_27a^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd~\\
A\_27a)~V0f1)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V1a)~V2a0)~V3f)~V4f0))=(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V1a)~(ap~V0f1~V2a0))~V3f)~V4f0)))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfbdpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f1 \in (((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  \\
({\unicdcdadad} V1a \in A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd~\\
A\_27a)~V0f1)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V1a)~V2a0)~V3f)~V4f0))=(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V1a)~V2a0)~(ap~V0f1~V3f))~V4f0)))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfafpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f1 \in (((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  \\
({\unicdcdadad} V1a \in A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd~\\
A\_27a)~V0f1)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V1a)~V2a0)~V3f)~V4f0))=(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V1a)~V2a0)~V3f)~(ap~V0f1~V4f0))))))))\end{array}
\label{hyp:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfdfcfnepfpfggfhaheg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedhfgngjgpfpfchjgoghgpfcdfedhfgngjgpfpfchjgoghgpfpfggogpfpffhahegbgehfgdh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0f1 \in (A\_27a^{{A\_27a}}) .  (\\
{\unicdcdadad} V1a \in A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR0\_\_fupd~\\
A\_27a)~V0f1)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V1a)~V2a0)~V3f)~V4f0))=(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~(ap~V0f1~V1a))~V2a0)~V3f)~V4f0))))))){\unicdcdcdhd}(({\unicdcdadad} V5f1 \in (A\_27a^{{A\_27a}}) .  \\
({\unicdcdadad} V6a \in A\_27a .  ({\unicdcdadad} V7a0 \in A\_27a .  ({\unicdcdadad} V8f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V9f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SR1\_\_fupd~\\
A\_27a)~V5f1)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V6a)~V7a0)~V8f)~V9f0))=(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V6a)~(ap~V5f1~V7a0))~V8f)~V9f0))))))){\unicdcdcdhd}(({\unicdcdadad} V10f1 \in (\\
((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ({\unicdcdadad} V11a \in \\
A\_27a .  ({\unicdcdadad} V12a0 \in A\_27a .  ({\unicdcdadad} V13f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V14f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRP\_\_fupd~\\
A\_27a)~V10f1)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V11a)~V12a0)~V13f)~V14f0))=(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V11a)~V12a0)~(ap~V10f1~V13f))~V14f0))))))){\unicdcdcdhd}({\unicdcdadad} V15f1 \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ({\unicdcdadad} V16a \in \\
A\_27a .  ({\unicdcdadad} V17a0 \in A\_27a .  ({\unicdcdadad} V18f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V19f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ((ap~(ap~(c\_2Esemi\_\_ring\_2Esemi\_\_ring\_\_SRM\_\_fupd~\\
A\_27a)~V15f1)~(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V16a)~V17a0)~V18f)~V19f0))=(ap~(ap~(ap~(ap~(c\_2Esemi\_\_ring\_2Erecordtype\_2Esemi\_\_ring~\\
A\_27a)~V16a)~V17a0)~V18f)~(ap~V15f1~V19f0)))))))))))\end{array}$$
\end{theorem}
\end{document}
