\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\_2Erat\_2ERDIV\_\_MUL\_\_OUT (TMd3fQggiSThsYf7xPUjEEDhYD96oEq1QmU)}
\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}
Let $ ty\_2Efrac\_2Efrac : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Efrac\_2Efrac}}\label{hyp:ogfgpfehjhpfcdfeggchbgdgpfcdfeggchbgdg}
\end{equation}
Let $ ty\_2Erat\_2Erat : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erat\_2Erat}}\label{hyp:ogfgpfehjhpfcdfechbgehpfcdfechbgeh}
\end{equation}
Let $ c\_2Erat\_2Erep\_\_rat\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Erat\_2Erep\_\_rat\_\_CLASS} \in {({{({{2}^{{ty\_2Efrac\_2Efrac}}})}^{{ty\_2Erat\_2Erat}}})}}\label{hyp:ngfgngpfdgpfcdfechbgehpfcdfechfgahpfpfchbgehpfpfdemebedfdf}
\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: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:dgpfcdfechbgehpfcdfechfgahpfpfchbgeh}
We define $\mname{c\_2Erat\_2Erep\_\_rat}$ to be ${{\uniddcgcg} V0a \in {ty\_2Erat\_2Erat} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Efrac\_2Efrac}})}}~{({{{ap}~{c\_2Erat\_2Erep\_\_rat\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ ty\_2Einteger\_2Eint : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Einteger\_2Eint}}\label{hyp:ogfgpfehjhpfcdfejgogehfghgfgchpfcdfejgogeh}
\end{equation}
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\_2Efrac\_2Erep\_\_frac : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Efrac\_2Erep\_\_frac \in ((ty\_2Epair\_2Eprod~ty\_2Einteger\_2Eint~\\
ty\_2Einteger\_2Eint)^{{ty\_2Efrac\_2Efrac}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggchbgdgpfcdfechfgahpfpfggchbgdg}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\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\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfogngch}
We define $\mname{c\_2Efrac\_2Efrac\_\_nmr}$ to be ${{\uniddcgcg} V0f \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{c\_2Epair\_2EFST}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Efrac\_2Erep\_\_frac}}~{V0f}})}})}}$.
\end{definition}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
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: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}
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 $ 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: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\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\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}
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\_\_lt : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_lt \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfmgeh}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfmgeh}
We define $\mname{c\_2Einteger\_2Eint\_\_lt}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_lt}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}}}$.
\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}
\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}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfebecedf}
We define $\mname{c\_2Einteger\_2EABS}$ to be ${{\uniddcgcg} V0n \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V0n}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{V0n}})}})}}~{V0n}})}}$.
\end{definition}
Let $ c\_2Epair\_2ESND : {{\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\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfegogng}
We define $\mname{c\_2Efrac\_2Efrac\_\_dnm}$ to be ${{\uniddcgcg} V0f \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{c\_2Epair\_2ESND}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Efrac\_2Erep\_\_frac}}~{V0f}})}})}}$.
\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:dgpfcdfejgogehfeihehfgogdhjgpgogpfcdfedfheoe}
We define $\mname{c\_2EintExtension\_2ESGN}$ to be ${{\uniddcgcg} V0x \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Einteger\_2Eint}})}}~{V0x}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V0x}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfdhhgog}
We define $\mname{c\_2Efrac\_2Efrac\_\_sgn}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{\mname{c\_2EintExtension\_2ESGN}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}$.
\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 $ 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}
Let $ c\_2Efrac\_2Eabs\_\_frac : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Efrac\_2Eabs\_\_frac \in (ty\_2Efrac\_2Efrac^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggchbgdgpfcdfebgcgdhpfpfggchbgdg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfngjgoggh}
We define $\mname{c\_2Efrac\_2Efrac\_\_minv}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_sgn}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2EABS}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpffgbhfhjggh}
We define $\mname{c\_2Erat\_2Erat\_\_equiv}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V1f2}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}}}$.
\end{definition}
Let $ c\_2Erat\_2Eabs\_\_rat\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Erat\_2Eabs\_\_rat\_\_CLASS} \in {({{ty\_2Erat\_2Erat}^{{({{2}^{{ty\_2Efrac\_2Efrac}}})}}})}}\label{hyp:ngfgngpfdgpfcdfechbgehpfcdfebgcgdhpfpfchbgehpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfebgcgdhpfpfchbgeh}
We define $\mname{c\_2Erat\_2Eabs\_\_rat}$ to be ${{\uniddcgcg} V0r \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Erat\_2Eabs\_\_rat\_\_CLASS}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_equiv}}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfngjgoggh}
We define $\mname{c\_2Erat\_2Erat\_\_minv}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_minv}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfngfhmg}
We define $\mname{c\_2Efrac\_2Efrac\_\_mul}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V1f2}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfngfhmg}
We define $\mname{c\_2Erat\_2Erat\_\_mul}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {{\uniddcgcg} V1r2 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_mul}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V1r2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfegjggh}
We define $\mname{c\_2Efrac\_2Efrac\_\_div}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_mul}}}~{V0f1}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_minv}}}~{V1f2}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfegjggh}
We define $\mname{c\_2Erat\_2Erat\_\_div}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {{\uniddcgcg} V1r2 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_div}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V1r2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0a \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1b \in ty\_2Erat\_2Erat .  (\\
{\unicdcdadad} V2c \in ty\_2Erat\_2Erat .  ((ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~V0a)~\\
(ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~V1b)~V2c))=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~\\
(ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~V0a)~V1b))~V2c)))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfneffmepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
((ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~V0r1)~V1r2)=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~\\
V0r1)~(ap~\mname{c\_2Erat\_2Erat\_\_minv}~V1r2)))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfeejegfpfpfneffmenejeoegf}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0t}})}}})}}})}{\unicdbdjded}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeoepeefpfpfoepeef}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{p}~{V0A}})}{\unicdbdjdcd}{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfpfpfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
(((p~V0A){\unicdbdjdcd}False){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((\preoh (p~V0A)){\unicdcdcdid}(p~V1B))){\unicdbdjdcd}False){\unicdbdjded}\\
((p~V0A){\unicdbdjdcd}((\preoh (p~V1B)){\unicdbdjdcd}False)))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfepecfpfpfeeffbemedd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0A \in {2} .  {({{({{({\preoh{{({{p}~{V0A}})}}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{({{({{({{p}~{V0A}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfebeoeeepfpfjeoegfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjded}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((p~V1q){\unicdcdcdid}(p~V2r))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}((\preoh (\\
p~V2r)){\unicdcdcdid}(\preoh (p~V1q)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V0p)))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}\\
((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpffgbh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  (((p~V0p){\unicdbdjded}(\\
(p~V1q){\unicdbdjdcd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(p~V1q)){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}((\\
\preoh (p~V1q)){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfjgngah}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdbdjdcd}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({{p}~{V0p}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogjgbd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdbdjdcd}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V1q}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogjgcd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdcdcdid}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V0p}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogpgbd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{\unicdcdadad} V1q \in {2} .  {({{({\preoh{{({{({{p}~{V0p}})}{\unicdcdcdid}{({{p}~{V1q}})}})}}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V1q}})}}})}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogpgcd}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0p \in {2} .  {({{({\preoh{{({\preoh{{({{p}~{V0p}})}}})}}})}{\unicdbdjdcd}{({{p}~{V0p}})}})}})}\label{hyp:ehigngpfcdfedhbgehpfcdfeahehigpfpfogog}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechbgehpfcdfecfeejegfpfpfneffmepfpfpeffef}
$$\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
({\unicdcdadad} V2r3 \in ty\_2Erat\_2Erat .  ((ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~V0r1)~\\
(ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~V1r2)~V2r3))=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~\\
(ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~V0r1)~V1r2))~V2r3)))))\end{array}$$
\end{theorem}
\end{document}
