\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\_2EintegerRing\_2Eint\_\_is\_\_ring (TMZqp7KiCv45wYNTmYk21mKceCSjEievSGa)}
\maketitle
\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:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
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 $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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: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: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: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}
Let $ ty\_2Epair\_2Eprod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Epair\_2EABS\_\_prod : {{\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\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfad}
We define $\mname{c\_2Einteger\_2Etint\_\_0}$ to be ${({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpffgbh}
\end{equation}
Let $ ty\_2Einteger\_2Eint : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Einteger\_2Eint}}\label{hyp:ogfgpfehjhpfcdfejgogehfghgfgchpfcdfejgogeh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_ABS\_\_CLASS \in (ty\_2Einteger\_2Eint^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedf}
We define $\mname{c\_2Einteger\_2Eint\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})} .  {({{{ap}~{c\_2Einteger\_2Eint\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfad}
We define $\mname{c\_2Einteger\_2Eint\_\_0}$ to be ${({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{\mname{c\_2Einteger\_2Etint\_\_0}}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfbd}
We define $\mname{c\_2Einteger\_2Etint\_\_1}$ to be ${({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbd}
We define $\mname{c\_2Einteger\_2Eint\_\_1}$ to be ${({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{\mname{c\_2Einteger\_2Etint\_\_1}}})}$.
\end{definition}
Let $ c\_2Einteger\_2Eint\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeafpfpfdemebedfdf}
\end{equation}
\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:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeaf}
We define $\mname{c\_2Einteger\_2Eint\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_neg : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_neg \in ((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfogfghg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfogfghg}
We define $\mname{c\_2Einteger\_2Eint\_\_neg}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_neg}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Eint\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Einteger\_2Eint\_\_of\_\_num} \in {({{ty\_2Einteger\_2Eint}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfpgggpfpfogfhng}
\end{equation}
Let $ c\_2Einteger\_2Etint\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfbgegeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbgegeg}
We define $\mname{c\_2Einteger\_2Eint\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_add}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_mul \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfngfhmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfngfhmg}
We define $\mname{c\_2Einteger\_2Eint\_\_mul}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_mul}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\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\_2Erecordtype\_2Ering : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a \in ((((((ty\_2Ering\_2Ering~A\_27a)^{{({{A\_27a}^{{A\_27a}}})}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}})^{{A\_27a}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechfgdgpgchegehjhahfgpfcdfechjgoghg}
\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: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}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\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}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0y \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1x \in ty\_2Einteger\_2Eint .  \\
((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V1x)~V0y)=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
V0y)~V1x))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfbeeeeepfpfdfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0y \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1x \in ty\_2Einteger\_2Eint .  \\
((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V1x)~V0y)=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V0y)~V1x))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfneffmepfpfdfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0z \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
({\unicdcdadad} V2x \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
V2x)~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V1y)~V0z))=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V2x)~V1y))~V0z)))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfbeeeeepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0z \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
({\unicdcdadad} V2x \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V2x)~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V1y)~V0z))=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V2x)~V1y))~V0z)))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfneffmepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
{({{\mname{c\_2Einteger\_2Eint\_\_0}}={({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\mname{c\_2Einteger\_2Eint\_\_1}=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V0x)=V0x))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfbeeeeepfpfmejeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
V0x)~(ap~\mname{c\_2Einteger\_2Eint\_\_neg}~V0x))=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfbeeeeepfpfcfjeoegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~V0x)=V0x))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfneffmepfpfmejeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
({\unicdcdadad} V2z \in ty\_2Einteger\_2Eint .  ((ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V0x)~V1y))~V2z)=(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~V0x)~V2z))~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_mul}~\\
V1y)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfcfeejedfefcfjece}
\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}}) .  \\
({\unicdcdadad} V4f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_R0~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V0a)~V1a0)~\\
V2f)~V3f0)~V4f1))=V0a)))))){\unicdcdcdhd}(({\unicdcdadad} V5a \in A\_27a .  ({\unicdcdadad} V6a0 \in \\
A\_27a .  ({\unicdcdadad} V7f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V8f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V9f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_R1~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V5a)~V6a0)~\\
V7f)~V8f0)~V9f1))=V6a0)))))){\unicdcdcdhd}(({\unicdcdadad} V10a \in A\_27a .  ({\unicdcdadad} V11a0 \in \\
A\_27a .  ({\unicdcdadad} V12f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V13f0 \in ((\\
A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V14f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_RP~\\
A\_27a)~(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
V10a)~V11a0)~V12f)~V13f0)~V14f1))=V12f)))))){\unicdcdcdhd}(({\unicdcdadad} V15a \in \\
A\_27a .  ({\unicdcdadad} V16a0 \in A\_27a .  ({\unicdcdadad} V17f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V18f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V19f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(c\_2Ering\_2Ering\_\_RM~A\_27a)~(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a)~V15a)~V16a0)~V17f)~V18f0)~V19f1))=V18f0)))))){\unicdcdcdhd}({\unicdcdadad} V20a \in \\
A\_27a .  ({\unicdcdadad} V21a0 \in A\_27a .  ({\unicdcdadad} V22f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V23f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V24f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(c\_2Ering\_2Ering\_\_RN~A\_27a)~(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a)~V20a)~V21a0)~V22f)~V23f0)~V24f1))=V24f1))))))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfbgdgdgfgdhdhpgchdh}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfejgogehfghgfgchcfjgoghgpfcdfejgogehpfpfjgdhpfpfchjgoghg}
$$\begin{array}{c}(p~(ap~(\mname{c\_2Ering\_2Eis\_\_ring}~ty\_2Einteger\_2Eint)~(ap~(ap~(ap~(ap~\\
(ap~(c\_2Ering\_2Erecordtype\_2Ering~ty\_2Einteger\_2Eint)~\mname{c\_2Einteger\_2Eint\_\_0})~\\
\mname{c\_2Einteger\_2Eint\_\_1})~\mname{c\_2Einteger\_2Eint\_\_add})~\mname{c\_2Einteger\_2Eint\_\_mul})~\\
\mname{c\_2Einteger\_2Eint\_\_neg})))\end{array}$$
\end{theorem}
\end{document}
