\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\_2Ering\_2Eplus\_\_zero\_\_right (TMQpN8Vp5AZrSMw6eSJSnKW8JjWdxa6iMJm)}
\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}
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\_\_RN : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RN~A\_27a \in ((A\_27a^{{A\_27a}})^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfoe}
\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\_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}
\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}
\begin{definition}\label{def:dgpfcdfechjgoghgpfcdfejgdhpfpfchjgoghg}
We define $\mname{c\_2Ering\_2Eis\_\_ring}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0r \in {({{ty\_2Ering\_2Ering}~{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\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}}~{V1n}})}}~{V2m}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RP}~{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\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}}~{V3n}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}}~{V4m}})}}~{V5p}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RP}~{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\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}}~{V6n}})}}~{V7m}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RM}~{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\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}}~{V8n}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}}~{V9m}})}}~{V10p}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RM}~{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\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_R0}~{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\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_R1}~{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\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}}~{V13n}})}}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RN}~{A\_27a}})}}~{V0r}})}}~{V13n}})}})}})}}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_R0}~{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\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}}~{V14n}})}}~{V15m}})}})}}~{V16p}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RP}~{A\_27a}})}}~{V0r}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}}~{V14n}})}}~{V16p}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ering\_2Ering\_\_RM}~{A\_27a}})}}~{V0r}})}}~{V15m}})}}~{V16p}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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}
\begin{theorem}\label{thm:ehigngpfcdfechjgoghgpfcdfeahmgfhdhpfpfkhfgchpgpfpfchjghgigeh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0r \in (ty\_2Ering\_2Ering~\\
A\_27a) .  ((p~(ap~(\mname{c\_2Ering\_2Eis\_\_ring}~A\_27a)~V0r)){\unicdbdjdcd}({\unicdcdadad} V1n \in \\
A\_27a .  ((ap~(ap~(ap~(c\_2Ering\_2Ering\_\_RP~A\_27a)~V0r)~V1n)~(ap~(c\_2Ering\_2Ering\_\_R0~\\
A\_27a)~V0r))=V1n))))\end{array}$$
\end{theorem}
\end{document}
