\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\_2ERATN\_\_DIV\_\_RATD (TMG6Yk2abhG6q2iHLnZrrA8juRTtXCjRioq)}
\maketitle
\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: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:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\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\_2Earithmetic\_2EEVEN : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EEVEN} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefegffeoe}
\end{equation}
Let $ c\_2Earithmetic\_2EODD : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EODD} \in {({{2}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepeeeee}
\end{equation}
\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}
\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: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}
\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 $ 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}
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: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}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfe}
We define $\mname{c\_2Earithmetic\_2E\_3E}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V1n}})}}~{V0m}})}}}$.
\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:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3E\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3E}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\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}
\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: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:dgpfcdfeahchjgngpfpfchfgdgpfcdfeafcffe}
We define $\mname{c\_2Eprim\_\_rec\_2EPRE}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Enum\_2Enum}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{\mname{c\_2Enum\_2E0}}})}})}}~{\mname{c\_2Enum\_2E0}}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V1n}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2EEXP : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EEXP \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefeifaf}
\end{equation}
Let $ c\_2Earithmetic\_2E\_2D : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2D \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdee}
\end{equation}
Let $ c\_2Earithmetic\_2E\_2A : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2A \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdbe}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfcdfejgkf}
We define $\mname{c\_2Enumeral\_2EiZ}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\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:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefcd}
We define $\mname{c\_2Earithmetic\_2EBIT2}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfdddepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3C\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\end{definition}
Let $ ty\_2Erat\_2Erat : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erat\_2Erat}}\label{hyp:ogfgpfehjhpfcdfechbgehpfcdfechbgeh}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
\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}
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\_\_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 $ ty\_2Efrac\_2Efrac : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Efrac\_2Efrac}}\label{hyp:ogfgpfehjhpfcdfeggchbgdgpfcdfeggchbgdg}
\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: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\_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\_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}
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 $ 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\_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:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfbgjgoggh}
We define $\mname{c\_2Efrac\_2Efrac\_\_ainv}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}}$.
\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\_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}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfbgegeg}
We define $\mname{c\_2Efrac\_2Efrac\_\_add}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_add}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_nmr}}}~{V1f2}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_mul}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V0f1}})}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_dnm}}}~{V1f2}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfdhfhcg}
We define $\mname{c\_2Efrac\_2Efrac\_\_sub}$ to be ${{\uniddcgcg} V0f1 \in {ty\_2Efrac\_2Efrac} .  {{\uniddcgcg} V1f2 \in {ty\_2Efrac\_2Efrac} .  {({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_add}}}~{V0f1}})}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_ainv}}}~{V1f2}})}})}}}$.
\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:dgpfcdfechbgehpfcdfechbgehpfpfdhfhcg}
We define $\mname{c\_2Erat\_2Erat\_\_sub}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {{\uniddcgcg} V1r2 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_sub}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V1r2}})}})}})}}}$.
\end{definition}
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: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}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfdhhgog}
We define $\mname{c\_2Erat\_2Erat\_\_sgn}$ to be ${{\uniddcgcg} V0r \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_sgn}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfmgfgdh}
We define $\mname{c\_2Erat\_2Erat\_\_les}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {{\uniddcgcg} V1r2 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_sgn}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_sub}}}~{V1r2}})}}~{V0r1}})}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfmgfgbh}
We define $\mname{c\_2Erat\_2Erat\_\_leq}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {{\uniddcgcg} V1r2 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_les}}}~{V0r1}})}}~{V1r2}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Erat\_2Erat}})}}~{V0r1}})}}~{V1r2}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfbgegeg}
We define $\mname{c\_2Erat\_2Erat\_\_add}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {{\uniddcgcg} V1r2 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_add}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V1r2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfad}
We define $\mname{c\_2Efrac\_2Efrac\_\_0}$ to be ${({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfad}
We define $\mname{c\_2Erat\_2Erat\_\_0}$ to be ${({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{\mname{c\_2Efrac\_2Efrac\_\_0}}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggchbgdgpfcdfeggchbgdgpfpfbd}
We define $\mname{c\_2Efrac\_2Efrac\_\_1}$ to be ${({{{ap}~{c\_2Efrac\_2Eabs\_\_frac}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Einteger\_2Eint}}~{ty\_2Einteger\_2Eint}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT1}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfbd}
We define $\mname{c\_2Erat\_2Erat\_\_1}$ to be ${({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{\mname{c\_2Efrac\_2Efrac\_\_1}}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfeoefhng}
We define $\mname{c\_2Einteger\_2ENum}$ to be ${{\uniddcgcg} V0i \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Einteger\_2Eint}})}}~{V0i}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{V1n}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfbgjgoggh}
We define $\mname{c\_2Erat\_2Erat\_\_ainv}$ to be ${{\uniddcgcg} V0r1 \in {ty\_2Erat\_2Erat} .  {({{{ap}~{\mname{c\_2Erat\_2Eabs\_\_rat}}}~{({{{ap}~{\mname{c\_2Efrac\_2Efrac\_\_ainv}}}~{({{{ap}~{\mname{c\_2Erat\_2Erep\_\_rat}}}~{V0r1}})}})}})}}$.
\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}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfmgfg}
We define $\mname{c\_2Einteger\_2Eint\_\_le}$ to be ${{\uniddcgcg} V0x \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1y \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V1y}})}}~{V0x}})}})}}}$.
\end{definition}
\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: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: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}
Let $ c\_2Erat\_2ERATN : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Erat\_2ERATN} \in {({{ty\_2Einteger\_2Eint}^{{ty\_2Erat\_2Erat}}})}}\label{hyp:ngfgngpfdgpfcdfechbgehpfcdfecfbeefoe}
\end{equation}
\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}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfedf}
We define $\mname{c\_2Ecombin\_2ES}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1g \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2x}})}}~{({{{ap}~{V1g}}~{V2x}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfeje}
We define $\mname{c\_2Ecombin\_2EI}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{ap}~{({{{{\mname{c\_2Ecombin\_2ES}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}}~{A\_27a}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{({{A\_27a}^{{A\_27a}}})}})}})}}~{({{{\mname{c\_2Ecombin\_2EK}}~{A\_27a}}~{A\_27a}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2Enum\_\_CASE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Earithmetic\_2Enum\_\_CASE~A\_27a \in \\
(((A\_27a^{{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})^{{A\_27a}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfeogfhngpfpfdebedffe}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfge}
We define $\mname{c\_2Erelation\_2EWF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1B \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2w \in {A\_27a} .  {({{{ap}~{V1B}}~{V2w}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3min \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1B}}~{V3min}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4b}})}}~{V3min}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1B}}~{V4b}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
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:dgpfcdfechfgmgbgehjgpgogpfcdfecffedfefcfjedeef}
We define $\mname{c\_2Erelation\_2ERESTRICT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{V1R}}~{V3y}})}}~{V2x}})}})}}~{({{{ap}~{V0f}}~{V3y}})}})}}~{({{c\_2Ebool\_2EARB}~{A\_27b}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeefde}
We define $\mname{c\_2Erelation\_2ETC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1a \in {A\_27a} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{({{2}^{{A\_27a}}})}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3P \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{V0R}}~{V4x}})}}~{V5y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V6x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V7y \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V8z \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V7y}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V7y}})}}~{V8z}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V6x}})}}~{V8z}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{V3P}}~{V1a}})}}~{V2b}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfebgahahchpgih}
We define $\mname{c\_2Erelation\_2Eapprox}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{A\_27b}^{{A\_27a}}})}})}}~{V3f}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V4y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4y}})}})}}~{V4y}})}})}})}}~{V0R}})}}~{V2x}})}})}}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfeehigfgpfpfggfhog}
We define $\mname{c\_2Erelation\_2Ethe\_\_fun}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Eapprox}}~{A\_27a}}~{A\_27b}})}}~{V0R}})}}~{V1M}})}}~{V2x}})}}~{V3f}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgmgbgehjgpgogpfcdfehfgecffede}
We define $\mname{c\_2Erelation\_2EWFREC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0R \in {({{({{2}^{{A\_27a}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1M \in {({{({{A\_27b}^{{A\_27a}}})}^{{({{A\_27b}^{{A\_27a}}})}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2Ethe\_\_fun}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Erelation\_2ETC}}~{A\_27a}})}}~{V0R}})}})}}~{({{\uniddcgcg} V3f \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V4v \in {A\_27a} .  {({{{ap}~{({{{ap}~{V1M}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2ERESTRICT}}~{A\_27a}}~{A\_27b}})}}~{V3f}})}}~{V0R}})}}~{V4v}})}})}}~{V4v}})}})}})}})}}~{V2x}})}})}}~{V0R}})}}~{V2x}})}})}}~{V2x}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfpgggpfpfogfhng}
We define $\mname{c\_2Erat\_2Erat\_\_of\_\_num}$ to be ${({{{ap}~{({{{ap}~{({{{\mname{c\_2Erelation\_2EWFREC}}~{ty\_2Enum\_2Enum}}~{ty\_2Erat\_2Erat}})}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{({{2}^{{ty\_2Enum\_2Enum}}})}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V0R \in {({{({{2}^{{ty\_2Enum\_2Enum}}})}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Erelation\_2EWF}}~{ty\_2Enum\_2Enum}})}}~{V0R}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{V0R}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V1n}})}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V1n}})}})}})}})}})}})}})}})}})}}~{({{\uniddcgcg} V2rat\_\_of\_\_num \in {({{ty\_2Erat\_2Erat}^{{ty\_2Enum\_2Enum}}})} .  {({{\uniddcgcg} V3a \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Earithmetic\_2Enum\_\_CASE}~{ty\_2Erat\_2Erat}})}}~{V3a}})}}~{({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{ty\_2Erat\_2Erat}})}}~{\mname{c\_2Erat\_2Erat\_\_0}}})}})}}~{({{\uniddcgcg} V4v \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Earithmetic\_2Enum\_\_CASE}~{ty\_2Erat\_2Erat}})}}~{V4v}})}}~{({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{ty\_2Erat\_2Erat}})}}~{\mname{c\_2Erat\_2Erat\_\_1}}})}})}}~{({{\uniddcgcg} V5n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ecombin\_2EI}}~{ty\_2Erat\_2Erat}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_add}}}~{({{{ap}~{V2rat\_\_of\_\_num}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V5n}})}})}})}}~{\mname{c\_2Erat\_2Erat\_\_1}}})}})}})}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechbgehpfcdfechbgehpfpfpgggpfpfjgogeh}
We define $\mname{c\_2Erat\_2Erat\_\_of\_\_int}$ to be ${{\uniddcgcg} V0i \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Erat\_2Erat}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_lt}}}~{V0i}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_ainv}}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_of\_\_num}}}~{({{{ap}~{\mname{c\_2Einteger\_2ENum}}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_neg}}}~{V0i}})}})}})}})}})}}~{({{{ap}~{\mname{c\_2Erat\_2Erat\_\_of\_\_num}}}~{({{{ap}~{\mname{c\_2Einteger\_2ENum}}}~{V0i}})}})}})}}$.
\end{definition}
Let $ c\_2Erat\_2ERATD : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Erat\_2ERATD} \in {({{ty\_2Enum\_2Enum}^{{ty\_2Erat\_2Erat}}})}}\label{hyp:ngfgngpfdgpfcdfechbgehpfcdfecfbeefee}
\end{equation}
\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}
Assume the following.
\begin{equation}
\begin{array}{c}((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO}))=\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeefhfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~(ap~c\_2Earithmetic\_2E\_2B~\mname{c\_2Enum\_2E0})~V0m)=V0m){\unicdcdcdhd}(((ap~(\\
ap~c\_2Earithmetic\_2E\_2B~V0m)~\mname{c\_2Enum\_2E0})=V0m){\unicdcdcdhd}(((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0m))~V1n)=(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0m)~V1n))){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n))=(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((\preoh (V0n=\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(\\
ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~V0n))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepeefpfpfkffecfpepfpfmeefpfpfkffecfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
\mname{c\_2Enum\_2E0})~V0n)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpepfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~(ap~c\_2Earithmetic\_2E\_2A~\mname{c\_2Enum\_2E0})~V0m)=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~V0m)=V0m){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))=V0m){\unicdcdcdhd}(\\
((ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Enum\_2ESUC}~V0m))~V1n)=(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~(ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~V1n))~\\
V1n)){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~(ap~\mname{c\_2Enum\_2ESUC}~V1n))=\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V0m)~V1n))))))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeneffmeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  (((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~V1n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1n)~V2p))){\unicdbdjdcd}(p~(\\
ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(V0m=V1n){\unicdbdjded}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V1n)){\unicdcdcdhd}(p~(\\
ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1n)~V0m))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfefebfpfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0m)~V2p))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1n)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfnepeoepepfpfmefedfdfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Enum\_2ESUC}~V0n)=(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))~V0n)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffdepfpfpeoefepfpfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({{p}~{V0t}})}{\unicdcdcdid}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifdemeffeefeeepfpfnejeeeeemefe}
\end{equation}
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}
{({{\unicdcdadad} V0t \in {2} .  {({{({{({{p}~{V0t}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfge}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({\preoh{{({{p}~{V0t}})}}})}{\unicdbdjdcd}{({{({{p}~{V0t}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepfpfjeneaf}
\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} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\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} 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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ((\preoh ({\unicdcdadad} V1x \in \\
A\_27a .  (p~(ap~V0P~V1x)))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27a .  (\preoh (p~(ap~V0P~V2x))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfgepecfbememepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  (((p~V0A){\unicdcdcdid}(\\
(p~V1B){\unicdcdcdid}(p~V2C))){\unicdbdjded}(((p~V0A){\unicdcdcdid}(p~V1B)){\unicdcdcdid}(p~V2C))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeejedfkepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((p~V0A){\unicdcdcdid}(p~V1B)){\unicdbdjded}((p~V1B){\unicdcdcdid}\\
(p~V0A)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeejedfkepfpfdfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdhd}(p~V1B))){\unicdbdjded}((\preoh (\\
p~V0A)){\unicdcdcdid}(\preoh (p~V1B)))){\unicdcdcdhd}((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjded}((\preoh (p~V0A)){\unicdcdcdhd}(\preoh (p~V1B)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeefepfpfnepecfhebeoepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{({{({{p}~{V0t}})}{\unicdbdjdcd}{False}})}{\unicdbdjded}{({{({{p}~{V0t}})}{\unicdbdjded}{False}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfgepfpffebfpfpfge}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjded}(p~V1t2)){\unicdbdjded}(((p~\\
V0t1){\unicdbdjdcd}(p~V1t2)){\unicdcdcdhd}((p~V1t2){\unicdbdjdcd}(p~V0t1))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfjeneafpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1x\_27 \in 2 .  ({\unicdcdadad} V2y \in 2 .  ({\unicdcdadad} V3y\_27 \in \\
2 .  ((((p~V0x){\unicdbdjded}(p~V1x\_27)){\unicdcdcdhd}((p~V1x\_27){\unicdbdjdcd}((p~V2y){\unicdbdjded}(p~V3y\_27)))){\unicdbdjdcd}\\
(((p~V0x){\unicdbdjdcd}(p~V2y)){\unicdbdjded}((p~V1x\_27){\unicdbdjdcd}(p~V3y\_27))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in 2 .  ({\unicdcdadad} V1Q \in 2 .  \\
({\unicdcdadad} V2x \in A\_27a .  ({\unicdcdadad} V3x\_27 \in A\_27a .  ({\unicdcdadad} V4y \in A\_27a .  \\
({\unicdcdadad} V5y\_27 \in A\_27a .  ((((p~V0P){\unicdbdjded}(p~V1Q)){\unicdcdcdhd}(((p~V1Q){\unicdbdjdcd}(V2x=V3x\_27)){\unicdcdcdhd}\\
((\preoh (p~V1Q)){\unicdbdjdcd}(V4y=V5y\_27)))){\unicdbdjdcd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\\
V0P)~V2x)~V4y)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~V1Q)~V3x\_27)~\\
V5y\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  ((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1))){\unicdcdcdhd}({\unicdcdadad} V2t1 \in A\_27a .  ({\unicdcdadad} V3t2 \in A\_27a .  ((ap~\\
(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~V2t1)~V3t2)=V3t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecgpgpgmgpfpfdgbgdhfgpfpfehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
((\preoh (p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~V0x)~V1y))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~\\
V1y)~V0x)))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfoepeefpfpfmeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V1y \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_lt}~V0x)~V1y)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~\\
V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfmeefpfpfjeneafpfpfmefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~(ap~\mname{c\_2Einteger\_2Eint\_\_neg}~\\
V0x))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~V0x)~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfoefehepfpfhefead}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Einteger\_2Eint .  (((ap~\mname{c\_2Einteger\_2Eint\_\_neg}~\\
V0x)=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0})){\unicdbdjded}(V0x=(ap~\\
c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfoefehepfpffebfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0i \in ty\_2Einteger\_2Eint .  (((ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
(ap~\mname{c\_2Einteger\_2ENum}~V0i))=V0i){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V0i))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpfpegepfpfoeffne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
(((ap~c\_2Einteger\_2Eint\_\_of\_\_num~V0m)=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
V1n)){\unicdbdjded}(V0m=V1n)))){\unicdcdcdhd}(({\unicdcdadad} V2x \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V3y \in \\
ty\_2Einteger\_2Eint .  (((ap~\mname{c\_2Einteger\_2Eint\_\_neg}~V2x)=(ap~\mname{c\_2Einteger\_2Eint\_\_neg}~\\
V3y)){\unicdbdjded}(V2x=V3y)))){\unicdcdcdhd}({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V5m \in \\
ty\_2Enum\_2Enum .  ((((ap~c\_2Einteger\_2Eint\_\_of\_\_num~V4n)=(ap~\\
\mname{c\_2Einteger\_2Eint\_\_neg}~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~V5m))){\unicdbdjded}\\
((V4n=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(V5m=\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(((ap~\mname{c\_2Einteger\_2Eint\_\_neg}~\\
(ap~c\_2Einteger\_2Eint\_\_of\_\_num~V4n))=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
V5m)){\unicdbdjded}((V4n=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(V5m=\mname{c\_2Enum\_2E0}))))))))\end{array}
\label{hyp:ehigngpfcdfejgogehfghgfgchpfcdfejeoeefpfpffebfpfpfdebemedeffmebeeffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\preoh ((ap~\mname{c\_2Enum\_2ESUC}~V0n)=\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfeogfhngpfcdfeoepeefpfpfdfffde}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0P \in (2^{{ty\_2Enum\_2Enum}}) .  (((p~(ap~V0P~\mname{c\_2Enum\_2E0})){\unicdcdcdhd}\\
({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((p~(ap~V0P~V1n)){\unicdbdjdcd}(p~(ap~V0P~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n)))))){\unicdbdjdcd}({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  (p~(ap~V0P~V2n)))))\end{array}
\label{hyp:ehigngpfcdfeogfhngpfcdfejeoeeeffdeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
\mname{c\_2Enum\_2E0})~V0n)=V0n)){\unicdcdcdhd}(({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V1n)~\mname{c\_2Enum\_2E0})=V1n)){\unicdcdcdhd}(({\unicdcdadad} V2n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V3m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V2n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V3m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Enumeral\_2EiZ}~(ap~\\
(ap~c\_2Earithmetic\_2E\_2B~V2n)~V3m)))))){\unicdcdcdhd}(({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2E\_2A~\mname{c\_2Enum\_2E0})~V4n)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}\\
(({\unicdcdadad} V5n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V5n)~\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V6n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V7m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~(\\
ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V6n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V7m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V6n)~V7m))))){\unicdcdcdhd}(({\unicdcdadad} V8n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
\mname{c\_2Enum\_2E0})~V8n)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V9n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2E\_2D~V9n)~\mname{c\_2Enum\_2E0})=V9n)){\unicdcdcdhd}(({\unicdcdadad} V10n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V11m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V10n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V11m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V10n)~V11m))))){\unicdcdcdhd}(({\unicdcdadad} V12n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~\\
\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V12n)))=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V13n \in ty\_2Enum\_2Enum .  ((ap~\\
(ap~c\_2Earithmetic\_2EEXP~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V13n)))=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}(({\unicdcdadad} V14n \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~V14n)~\mname{c\_2Enum\_2E0})=\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}\\
(({\unicdcdadad} V15n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V16m \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2EEXP~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V15n))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V16m))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~(ap~c\_2Earithmetic\_2EEXP~V15n)~V16m))))){\unicdcdcdhd}(((ap~\mname{c\_2Enum\_2ESUC}~\\
\mname{c\_2Enum\_2E0})=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V17n \in ty\_2Enum\_2Enum .  (\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V17n))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V17n)))){\unicdcdcdhd}(((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\mname{c\_2Enum\_2E0})=\\
\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(({\unicdcdadad} V18n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V18n))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Eprim\_\_rec\_2EPRE}~V18n)))){\unicdcdcdhd}(({\unicdcdadad} V19n \in ty\_2Enum\_2Enum .  \\
(((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V19n)=\mname{c\_2Enum\_2E0}){\unicdbdjded}(V19n=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}\\
(({\unicdcdadad} V20n \in ty\_2Enum\_2Enum .  ((\mname{c\_2Enum\_2E0}=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V20n)){\unicdbdjded}(V20n=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V21n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V22m \in ty\_2Enum\_2Enum .  (((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V21n)=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V22m)){\unicdbdjded}(V21n=V22m)))){\unicdcdcdhd}\\
(({\unicdcdadad} V23n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V23n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}False)){\unicdcdcdhd}(({\unicdcdadad} V24n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V24n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Earithmetic\_2EZERO})~\\
V24n)))){\unicdcdcdhd}(({\unicdcdadad} V25n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V26m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V25n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V26m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V25n)~V26m))))){\unicdcdcdhd}(({\unicdcdadad} V27n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~\\
\mname{c\_2Enum\_2E0})~V27n)){\unicdbdjded}False)){\unicdcdcdhd}(({\unicdcdadad} V28n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V28n))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Earithmetic\_2EZERO})~\\
V28n)))){\unicdcdcdhd}(({\unicdcdadad} V29n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V30m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V29n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V30m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V30m)~V29n))))){\unicdcdcdhd}(({\unicdcdadad} V31n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
\mname{c\_2Enum\_2E0})~V31n)){\unicdbdjded}True)){\unicdcdcdhd}(({\unicdcdadad} V32n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V32n))~\mname{c\_2Enum\_2E0})){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V32n)~\\
\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}(({\unicdcdadad} V33n \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V34m \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V33n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V34m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V33n)~V34m))))){\unicdcdcdhd}(\\
({\unicdcdadad} V35n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\\
V35n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}True)){\unicdcdcdhd}(({\unicdcdadad} V36n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\mname{c\_2Enum\_2E0})~V36n)){\unicdbdjded}(V36n=\\
\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(({\unicdcdadad} V37n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V38m \in \\
ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V37n))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V38m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V38m)~V37n))))){\unicdcdcdhd}(({\unicdcdadad} V39n \in ty\_2Enum\_2Enum .  ((p~(ap~c\_2Earithmetic\_2EODD~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V39n))){\unicdbdjded}(p~(ap~c\_2Earithmetic\_2EODD~\\
V39n)))){\unicdcdcdhd}(({\unicdcdadad} V40n \in ty\_2Enum\_2Enum .  ((p~(ap~c\_2Earithmetic\_2EEVEN~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V40n))){\unicdbdjded}(p~(ap~c\_2Earithmetic\_2EEVEN~\\
V40n)))){\unicdcdcdhd}((\preoh (p~(ap~c\_2Earithmetic\_2EODD~\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(p~(\\
ap~c\_2Earithmetic\_2EEVEN~\mname{c\_2Enum\_2E0})))))))))))))))))))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfegjgdhehchjgcg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  (\\
((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\mname{c\_2Earithmetic\_2EZERO})~V0n)){\unicdbdjded}\\
True){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~\mname{c\_2Earithmetic\_2EZERO})){\unicdbdjded}False){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V0n))~\mname{c\_2Earithmetic\_2EZERO})){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m))){\unicdcdcdhd}(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m))){\unicdbdjded}(\preoh (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1m)~V0n)))){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~V1m)))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfmgehfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~\mname{c\_2Enum\_2ESUC}~V0m)=(ap~\mname{c\_2Enum\_2ESUC}~V1n)){\unicdbdjded}(V0m=V1n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfejeoegfpfpfdfffdepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0a \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1b \in ty\_2Erat\_2Erat .  (\\
(ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~V0a)~V1b)=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~\\
V1b)~V0a))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfbeeeeepfpfdepenene}
\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\_\_sub}~V0r1)~V1r2)=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~\\
V0r1)~(ap~\mname{c\_2Erat\_2Erat\_\_ainv}~V1r2)))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfdfffcepfpfbeeeeebejeoegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((ap~\mname{c\_2Erat\_2Erat\_\_ainv}~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\mname{c\_2Enum\_2E0}))=\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\mname{c\_2Enum\_2E0}))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfbejeoegfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
((ap~\mname{c\_2Erat\_2Erat\_\_ainv}~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~V0r1)~V1r2))=\\
(ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~(ap~\mname{c\_2Erat\_2Erat\_\_ainv}~V0r1))~(ap~\\
\mname{c\_2Erat\_2Erat\_\_ainv}~V1r2)))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfbejeoegfpfpfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
(((ap~\mname{c\_2Erat\_2Erat\_\_ainv}~V0r1)=V1r2){\unicdbdjded}(V0r1=(ap~\mname{c\_2Erat\_2Erat\_\_ainv}~\\
V1r2)))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfbejeoegfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
(((ap~\mname{c\_2Erat\_2Erat\_\_ainv}~V0r1)=(ap~\mname{c\_2Erat\_2Erat\_\_ainv}~V1r2)){\unicdbdjded}\\
(V0r1=V1r2))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpffebfpfpfbejeoegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
((p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_les}~V0r1)~V1r2)){\unicdbdjdcd}(\preoh (p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_les}~\\
V1r2)~V0r1))))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfmefedfpfpfbeoeefjedfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  (p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_leq}~V0r1)~\\
V0r1)))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfmefebfpfpfcffege}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_les}~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\mname{c\_2Enum\_2E0}))~\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\\
\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfmefedfpfpfadbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
((p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_les}~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\mname{c\_2Enum\_2E0}))~\\
V0r1)){\unicdbdjdcd}((p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_leq}~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\\
\mname{c\_2Enum\_2E0}))~V1r2)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_les}~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\\
\mname{c\_2Enum\_2E0}))~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~V0r1)~V1r2)))))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfadmefedfpfpfadmefebfpfpfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
({\unicdcdadad} V2r3 \in ty\_2Erat\_2Erat .  ((V0r1=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_sub}~\\
V1r2)~V2r3)){\unicdbdjded}((ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~V0r1)~V2r3)=V1r2)))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfcfdfffcepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
({\unicdcdadad} V2r3 \in ty\_2Erat\_2Erat .  ((\preoh (V2r3=(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\\
\mname{c\_2Enum\_2E0}))){\unicdbdjdcd}((V0r1=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~V1r2)~V2r3)){\unicdbdjded}\\
((ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~V0r1)~V2r3)=V1r2))))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfcfeejegfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\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\_\_add}~V0r1)~\\
V2r3)=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~V1r2)~V2r3)){\unicdbdjded}(V0r1=V1r2)))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpffebfpfpfcfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r1 \in ty\_2Erat\_2Erat .  ({\unicdcdadad} V1r2 \in ty\_2Erat\_2Erat .  \\
((p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_les}~(ap~\mname{c\_2Erat\_2Erat\_\_ainv}~V0r1))~\\
(ap~\mname{c\_2Erat\_2Erat\_\_ainv}~V1r2))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Erat\_2Erat\_\_les}~\\
V1r2)~V0r1)))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfmefedfpfpfbejeoegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0n \in A\_27a .  (((ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\\
\mname{c\_2Enum\_2E0})=\mname{c\_2Erat\_2Erat\_\_0}){\unicdcdcdhd}({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
((ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~(ap~\mname{c\_2Enum\_2ESUC}~V1n))=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_add}~\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~V1n))~\mname{c\_2Erat\_2Erat\_\_1})))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfpegepfpfoeffne}
\end{equation}
Assume the following.
\begin{equation}
{({{\mname{c\_2Erat\_2Erat\_\_0}}={({{{ap}~{\mname{c\_2Erat\_2Erat\_\_of\_\_num}}}~{\mname{c\_2Enum\_2E0}}})}})}\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\mname{c\_2Erat\_2Erat\_\_1}=(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefpfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r \in ty\_2Erat\_2Erat .  ((V0r=(ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_int}~(ap~c\_2Erat\_2ERATN~V0r)))~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\\
(ap~c\_2Erat\_2ERATD~V0r)))){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
(ap~c\_2Erat\_2ERATD~V0r))){\unicdcdcdhd}((((ap~c\_2Erat\_2ERATN~V0r)=(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})){\unicdbdjdcd}((ap~c\_2Erat\_2ERATD~V0r)=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}({\unicdcdadad} V1n\_27 \in \\
ty\_2Einteger\_2Eint .  ({\unicdcdadad} V2d\_27 \in ty\_2Enum\_2Enum .  (((V0r=\\
(ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_int}~V1n\_27))~\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~V2d\_27))){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V2d\_27))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_le}~(ap~\mname{c\_2Einteger\_2EABS}~\\
(ap~c\_2Erat\_2ERATN~V0r)))~(ap~\mname{c\_2Einteger\_2EABS}~V1n\_27))))))))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefoeeepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r \in ty\_2Erat\_2Erat .  ((ap~(ap~\mname{c\_2Erat\_2Erat\_\_mul}~V0r)~\\
(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~(ap~c\_2Erat\_2ERATD~V0r)))=(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_int}~\\
(ap~c\_2Erat\_2ERATN~V0r))))\end{array}
\label{hyp:ehigngpfcdfechbgehpfcdfecfbeefoepfpfcfbeefeepfpfneffmeef}
\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){\unicdcdcdhd}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V2r)))){\unicdcdcdhd}(((p~V1q){\unicdcdcdid}\\
(\preoh (p~V0p))){\unicdcdcdhd}((p~V2r){\unicdcdcdid}(\preoh (p~V0p)))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfdgpgogkg}
\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){\unicdcdcdid}(p~V2r))){\unicdbdjded}(((p~V0p){\unicdcdcdid}(\preoh (p~V1q))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}(\preoh (p~V2r))){\unicdcdcdhd}\\
((p~V1q){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (p~V0p))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfegjgdhkg}
\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}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  (((p~V0p){\unicdbdjded}(\preoh (p~V1q))){\unicdbdjded}(((p~V0p){\unicdcdcdid}\\
(p~V1q)){\unicdcdcdhd}((\preoh (p~V1q)){\unicdcdcdid}(\preoh (p~V0p)))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfogfghg}
\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:ehigngpfcdfechbgehpfcdfecfbeefoepfpfeejegfpfpfcfbeefee}
$$\begin{array}{c}({\unicdcdadad} V0r \in ty\_2Erat\_2Erat .  ((ap~(ap~\mname{c\_2Erat\_2Erat\_\_div}~(ap~\\
\mname{c\_2Erat\_2Erat\_\_of\_\_int}~(ap~c\_2Erat\_2ERATN~V0r)))~(ap~\mname{c\_2Erat\_2Erat\_\_of\_\_num}~\\
(ap~c\_2Erat\_2ERATD~V0r)))=V0r))\end{array}$$
\end{theorem}
\end{document}
